Chapter 7 is a critical section because it concerns the introduction of a better priesthood. No sacrifices were possible without a priest (therefore no "forgiveness") and therefore the priesthood was greatly revered by Jews.

Remember that in Hebrews 5:5-10 the writer began to speak of the Melchizedek but then began a lengthy parenthetical section from Hebrews 5:11 through Hebrews 6:20. This parenthesis was to prepare his "dull of hearing" readers so that they might better understand of this important chapter. There is a point of application we in the modern church need to consider and it is this -- the solid food and sound doctrine of Scripture is not revealed by the Spirit to those who are spiritually lazy or apathetic. In Hebrews 7:4 we encounter the only command in this entire chapter (see note) and it is to give careful consideration (as contrasted with a superficial reading) to this teaching on Melchizedek because he is a type of Jesus Christ, the One Whose beauty and glory we desire to see.

The background of course is Jewish readers who had professed belief in Christ and were being tempted to abandon their faith in the Messiah and return to Judaism under threat of persecution as described in chapter 10...

But remember the former days, when, after being enlightened, you endured a great conflict of sufferings, 33 partly, by being made a public spectacle through reproaches and tribulations, and partly by becoming sharers with those who were so treated. 34 For you showed sympathy to the prisoners, and accepted joyfully the seizure of your property, knowing that you have for yourselves a better possession and an abiding one. (see notes Hebrews 10:30; 31; 32)

And so the writer is writing to convince them to break with the familiar system and rituals that their forefathers had followed for centuries. He is saying that the religious system of sacrifices, rituals, and rules that had been practice for some 1,400 years had now been replaced by a better way. And so he focuses on the supremacy of Christ Jesus the great High Priest, Who is the fulfillment of all that was written by Moses and the prophets. And so he picks up his thoughts on Melchizedek from Hebrews 5 because he wants to explain that Jesus is not a priest like the familiar Levitical priests, but is of a different order of Melchizedek and because of that, He is a better priest than any of the priests in the old system. Ironically the old system was never intended to be the end but the means, a means which was always and in all ways intended to point to the coming Messiah, Jesus Christ.

Ray Stedman points out that...

The unfolding of the meaning of the Melchizedek priesthood of Jesus is the goal toward which the author has been aiming ever since Hebrews 2:17 (note), where he first uses the term high priest with reference to Jesus... These themes are little noted or understood in the average church today but desperately needed if the church (or the individual Christian) is to confront the world with power and grace. (Stedman, Ray: Hebrews IVP New Testament Commentary Series or Logos) (Bolding added for emphasis)

As Matthew Henry says...

Nothing made the Jews so fond of the Levitical dispensation as the high esteem they had of their priesthood, and it was doubtless a sacred and most excellent institution; it was a very severe threatening denounced against the Jews (Hos. 3:4), that the children of Israel should abide many days without a prince or priest, and without a sacrifice, and with an ephod, and without teraphim. Now the apostle assures them that by receiving the Lord Jesus they would have a much better high priest, a priesthood of a higher order, and consequently a better dispensation or covenant, a better law and testament...(and now the writer) sets before them some of the strong meat he had spoken of before, hoping they would by greater diligence be better prepared to digest it.

Expositor's Greek Testament writes that...

The subject of Christ's priesthood is resumed; the interpolated admonition (Hebrews 5:11-6:20) having been skillfully brought round to a second mention (actually strictly speaking the 4th mention) of Melchizedek. The chief reason for introducing the priesthood of Melchizedek as the type of Christ's priesthood was that it was "for ever". The Aaronic priesthood was successional, this single; and in this sense "for ever". There were, however, other reasons. The first question with a Jew who was enjoined to trust Christ's priestly mediation, would be "What are His orders?" He belonged to a tribe of which Moses had spoken nothing concerning priesthood. He might or might not be the true heir to David's throne; but if He was, did not this very circumstance exclude Him from the priestly office? Was it credible that the nation had been encouraged rigorously to exclude from the priesthood every interloper, only in order that at last this rigidly preserved order should be entirely disregarded? This writer seizes upon the fact that there was a greater priest than Aaron mentioned in Scripture -- a priest more worthy to be the type of the Messianic priesthood, because He was Himself a King, and especially because He belonged to no successional priestly order but was Himself the entire order. (!) This idea of a priesthood superseding that of Levi's sons found its way into Scripture through the hymn (Psalm 110:4)... The chapter may be divided thus
 

I. Characteristics of Melchizedek, vv1-10

1. In himself as depicted in Scripture, vv1-3
2. In his relation to Levi and his line, vv4-10

II. Inadequacy of Levitical priesthood in comparison with the Melchizedek priesthood of Christ, vv11-25

1. Levi, being provisional, Melchizedek being permanent, vv11-14
2. Official and hereditary : personal and eternal, vv15-19
3. Without oath: with oath, therefore final, vv20-22
4. Plural and successional : singular and enduring, vv23-25

III. Summary of the merits of the new Melchizedek Priest, Jesus
(from Nicoll, W Robertson, Editor: Expositors Greek Testament: 5 Volumes. Out of print. Search Google)

John Piper helps us understand why all of this focus on Melchizedek and the priesthood, things which seem so foreign and even may seem unnecessary to us in the modern western culture. The writer is saying we are in desperate need of a High Priest for as Piper explains...

 the reason for all this talk about Christ’s relation to Melchizedek in verses 1–24 is because the eternal, superior priesthood of Jesus is our only hope of eternal salvation. God’s wrath never changes. There is only one hope for sinners like us. We must have a faithful High Priest, Who will intercede for us forever. We need a King of righteousness (Hebrews 7:2). We need a King of peace (Hebrews 7:2). We need Someone without beginning and ending (Hebrews 7:3). Someone Who has an indestructible life (verse 16) and will never die and need to be replaced (Hebrews 7:23, 24). We need Someone greater than Abraham and greater than Levi—something like Melchizedek, who blessed Abraham, (Hebrews 7:6; 7:7) and who received tithes from Abraham and, in a sense, from Levi in Abraham (Hebrews 7:5, 7:6, 7:8, 7:9, 7:10). We need a new and greater Priest—so much greater that verse 11 says there was no perfection through the Levitical priesthood. All the Old Testament priesthood could do was point toward the One superior Priest (after the order of Melchizedek, Psalm 110:4), Whose sacrifice of Himself and Whose eternal intercession would guarantee eternal salvation for all God’s people.

So the first implication of Hebrews 7:25 is that all this truth about priesthood is because what we need saving from is the wrath of God. God’s way of solving that problem is priesthood. This is not ours to figure out or solve. God has to do it for us. And He has done it. He ordains a Priest, His Son.

And don’t make a mistake here. It’s not as though Jesus the Priest loves us and God the Father doesn’t. God the Father ordains the priesthood for our salvation. It is His idea. He sends the Priest. It is His own Son Whom He sends. And He loves Him infinitely. All this is the love of God rescuing us from the wrath of God, in such a way that the justice of God is vindicated and the glory of God is exalted. (Read his entire excellent message on Hebrews 7:1-25 Jesus from Melchizedek to Savior)

FOR THIS MELCHIZEDEK: houtos gar ho Melchisedek: (Heb 6:20; Genesis 14:18-20 )

For - He is explaining the end of the previous verse...

Jesus has entered as a forerunner for us, having become a high priest forever according to the order of Melchizedek. (see note Hebrews 6:20)

Expositor's Greek Testament feels that...

it is the "for-everness" of the priesthood which he means especially to insist upon. The whole order is occupied by himself. This one man constitutes the order. He succeeds no one in office and no one succeeds him. In this sense he abides a priest for ever.  Between the subject Melchizedek and the verb "remains" there are inserted five historical facts taken from Genesis 14, with their interpretation.

Melchizedek - This is the writer's fourth mention of Melchizedek (see all 8 uses). In the 28 verses in Hebrews 7 the writer gives us what amounts to an exposition of the OT Scriptures on Melchizedek because there were only 3 verses in Genesis 14 and only one notation some 500 years later by David in Psalm 110:4.

Genesis 14:17 (Moses writes) Then after his return from the defeat of Chedorlaomer and the kings who were with him, the king of Sodom went out to meet him at the valley of Shaveh (that is, the King's Valley).
18 And Melchizedek (means "king of righteousness") king of Salem (means "peace", thus Melchizedek was "king of peace". Salem in this context is another name for Jerusalem and the first mention of the "city of God" in the Bible) brought out bread and wine; now he was a priest (the first mention of a priest in Scripture) of God Most High (El Elyon).
19 And he blessed him and said, "Blessed be Abram of God Most High, Possessor of heaven and earth;
20 And blessed be God Most High, Who has delivered your enemies into your hand." And he (Abraham) gave him (Melchizedek) a tenth of all.
21 And the king of Sodom said to Abram, "Give the people to me and take the goods for yourself."
22 And Abram said to the king of Sodom, "I have sworn to the LORD God Most High, possessor of heaven and earth,
23 that I will not take a thread or a sandal thong or anything that is yours, lest you should say, 'I have made Abram rich.'
24 "I will take nothing except what the young men have eaten, and the share of the men who went with me, Aner, Eshcol, and Mamre; let them take their share."

Psalm 110:4 (David writes that) The LORD has sworn and will not change His mind, "Thou art a priest forever according to the order of Melchizedek."

At the time of the writing of this epistle to the Hebrews another 1000 years had passed since David's declaration in Psalm 110:4. And so the writer of Hebrews begins to give more detail regarding Melchizedek and how this historical OT individual validates the priesthood of Jesus.

Dwight Pentecost has the following application writing that...

If a writer of the New Testament considered Old Testament episodes—even something so brief as this encounter with Melchizedek—so important, we would do well to become as familiar as possible with the wealth of information God has revealed in the Old Testament! (Pentecost, J. D., & Durham, K. . Faith that Endures. Grand Rapids, MI: Kregel Publications)

H A Ironside comments that...

Abraham recognized Melchisedec's spiritual authority by giving him tithes of all the spoils. Strengthened by the bread and wine administered by Salem's king-priest, Abraham was prepared to refuse the blandishments of the King of Sodom, representative of the world in all its impurity and debasement. (H. A. Ironside Expository Commentary on Hebrews)

Guzik comments on the subtle juxtaposition of Melchizedek, the king of righteousness followed by peace (the king of Salem or peace)...

As always, righteousness comes before peace. Righteousness is the only true path to peace. People look for that peace in escape, in evasion, or in compromise; but they will only find it in righteousness. (Hebrews 7)

Operation Entebbe is a fascinating modern day "equivalent" of the Genesis 14 account of Abraham's daring exploit. The following account of Operation Entebbe is taken from Wikipedia...

On June 27, 1976, Air France Flight 139, an Airbus A300 originating from Tel Aviv, carrying 248 passengers and a crew of twelve, took off from Athens, heading for Paris. Soon after the 12:30 p.m. takeoff, the flight was hijacked by two Palestinians from the Popular Front for the Liberation of Palestine - External Operations (PFLP-EO) and two Germans from the German "Revolutionary Cells (RZ)" (Wilfried Böse and Brigitte Kuhlmann), who commandeered the flight, diverting it to Benghazi, Libya. There it was held on the ground for seven hours for refueling, during which time a female hostage who pretended she was pregnant was released. The plane left Benghazi, and at 3:15 it arrived at Entebbe Airport in Uganda. (Read the  full account)

Uganda at that time was ruled by the dictator Idi Amin who welcomed the hijacked airliner which remained at Entebbe Airport the next 7 days as the PLO hijackers prepared for their next move. Like the raiders who had taken Lot captive in Genesis 14, the PLO hijackers certainly appeared to be in total control of their Israeli captives who were on this flight. But unbeknownst to the hijackers, three Israeli C-130 Hercules transports were on there way from Tel Aviv to Entebbe with a Israeli commandos, who within hours attacked the airport under the cover of darkness. In less than one hour the commandos rushed the old terminal, gunned down the hijackers, and rescued 110 of the 113 hostages. The next day Israel’s Premier Yitzhak Rabin declared "This operation will certainly be inscribed in the annals of military history, in legend and in national tradition” and indeed it has been so honored, even as was Abraham's daring raid some 4000 years earlier! Abraham with only “318 trained men” from his own household took off in hot pursuit of Lot and his kidnappers and under the cover of night deployed his relatively small force in a surprise attack which put the four kidnapper kings to flight.

So when Abraham returned to his home after the slaughter of the kings he was a hero, at the pinnacle of martial success. Can you see him proudly astride his lumbering camel, smeared with the dirt and blood of battle, leading his 318 proud men plus Lot and all the captives and all the plunder through Jerusalem? If so, you have the “feel” necessary to begin to appreciate Abraham’s strange, mystic encounter with a shadowy figure of immense grandeur — Melchizedek, the priest-king of Salem.

In this background and a sense of Abraham's incredible victory against all odds we read these words in Hebrews 7...

This Melchizedek, king of Salem, priest of the Most High God, who met Abraham as he was returning from the slaughter of the kings and blessed him

Imagine the joy and victory that was in the air. It would have been enough to go to any man's head, but not Abraham, for he recognized that the victory was not his, but ultimately that they victory was the Lord's. And so he chose to honor King Melchizedek who was also the priest of the Most High God Who had given him the victory. Our victories from day to day may not be as dramatic but they are no less always a reflection of the fact that the victory belongs to the Lord. Are we quick to acknowledge this eternal truth when accolades and adulation come our way? Let us seek to have an "Abraham attitude" as we bask in the victory, whether it be over our flesh, the world or the devil. As David said some 500 years later

Thine, O LORD, is the greatness and the power and the glory and the victory and the majesty, indeed everything that is in the heavens and the earth; Thine is the dominion, O LORD, and Thou dost exalt Thyself as head over all. (1 Chronicles 29:11)

As an aside, the reader will undoubtedly encounter a wide variety of interpretations on the identity of Melchizedek, some of the more fanciful interpretations including the following...

(1). Angel = Origen, Didymus

(2). Enoch = Husius, Calmet

(3). Shem = the rabbis of the first century sought to identify him with Shem, the oldest son of Noah, to counter the Christian view of him as a type of Christ. The early Christian writers for the most part objected to this as invalidating the claim of Hebrews that Melchizedek was “without genealogy,” since the genealogy of Shem was well known.

(4) Some Gnostic cults (Gnosticism) taught that Melchizedek was a theophany (visible manifestation) of the Holy Spirit, while a later sect saw him as a preincarnate appearance of the Son of God.

(5) The Dead Sea Scrolls  (Cave 11 at Qumran) spoke of Melchizedek as the coming great Deliverer of the Jewish remnant and equated him with the archangel Michael.

(6) Jerome (340-420 AD), author of the Latin Vulgate translation of the Bible , stated that early church authorities such as Irenaeus, Hippolytus, Eusebius of Caesarea and Apollinaris all viewed Melchizedek as a man and the majority of the Reformers followed this view.

A number of writers interpret Melchizedek as a preincarnate appearance of Christ and base this interpretation on facts from the OT and Hebrews...

(1). The names, the king of righteousness and king of peace are very similar to names attributed to Christ (Hebrews 7:2)

(2). His lack of a recorded genealogy - "without father, without mother, without genealogy, having neither beginning of days nor end of life" (Hebrews 7:3)

(3). "He abides a priest perpetually" (Hebrews 7:3)

(4). Contrasted with mortal men - "mortal men receive tithes, but in that case one (Melchizedek) receives them, of whom it is witnessed that he lives on" (Hebrews 7:8)

One of the strongest arguments against that interpretation is the writer's clear statement that he was "made like the Son of God" (see note Hebrews 7:3). See below for a synopsis of how many conservative (and one cult) sources interpret the identity of Melchizedek.

The Targum, which reflects rabbinic interpretation, Melchizedek is identified as Shem, Noah's son (Targum Ps.-J. Gen 14:18; see esp Carmona, Est Bib 37 [1978] 79-102). The rabbis made the totally unsubstantiated remark that the priesthood was transferred to Abraham and his posterity at the meeting recorded in Genesis 14:18-20 because Melchizedek proved to be unworthy of his office! The implications of this specious rabbinic interpretation would negate the writer's argument in Hebrews 7 in view of the fact that Levi eventually came from the line of Abraham and thus  the Levitical priesthood is the legitimate successor to the priesthood forfeited by Melchizedek. The rabbinic literature goes on to apply the description of Psalm 110:4 to Abraham (cp, R. Ishmael as early as 135 AD) .

Gerald Hawthorne writes that Melchizedek's...

greatness is seen from the fact that he blessed Abraham the patriarch (i.e., the father of us all) at a time when Abraham was second to none in the land—victor over Chedorlaomer and the kings who were with him, and from the fact that Abraham gave him a tenth of everything. (New International Bible Commentary)

KING OF SALEM, PRIEST OF THE MOST HIGH GOD: basileus Salem hiereus tou Theou tou hupsistou: (Psalms 76:2) (Psalms 57:2; 78:35,56; Daniel 4:2; 5:18,21; Micah 6:6; Mark 5:7; Acts 16:17)

King of Salem - In the next verse the writer tells us this equates with king of peace.

Salem is also another name for Jerusalem, as deduced from Psalm 76...

God is known in Judah; His name is great in Israel. And His tabernacle is in Salem; His dwelling place also is in Zion. (Ps 76:1,2)

The Hebrew word for ‘Salem” is shalem (08004) which is in turn derived from the same Hebrew root as shalom “peace”.  Genesis 14:18 and Psalm 76:2 are the only occurrences of Salem in the OT.

Priest (2409) (hiereus from hieros = sacred, holy, consecrated to God, used as a noun to mean a sacred place or temple, cp Mark 11:11) is a sacred or consecrated person who serves God (or the false gods in the pagan religions). Hiereus describes the specific position and not necessarily a priest’s character.

W. G. Moorehead defines a priest as

One who is duly qualified to minister in sacred things, particularly to offer sacrifices at the altar, and to act as mediator between men and God. (ISBE)

Priest in Latin is pontifex (from pont-, pons = bridge + facere = to make) which literally means a bridge maker, and is word used even today to describe the Pope as "Pontifex Maximus" (maximus = greatest, highest), which literally means the "greatest bridge builder" and in modern parlance "the Highest Priest".

Wuest comments that...

The Roman emperor was Pontifex Maximus, a high priest upon the throne of the Caesars. But our Lord Jesus is a high priest who, now seated upon a throne of grace, will some day as High Priest in the Messianic Kingdom occupy the throne of David in Jerusalem, as Zechariah says, “He shall be a priest upon his throne” (Zech. 6:13).

The Most High God - (See all NT uses of phrase "Most High") In Hebrew "Most High God" is El Elyon (note) one of the great names of God (they are all great of course!) which is expressive of God's sovereignty over all things.

In the Old Testament, the throne (KING) and the altar (PRIESTS) were separated and any person who was not of the Levitical priesthood who attempted to act as priest was judged by God.

But Uzziah (a king who was attempting to function as a priest), with a censer in his hand for burning incense, was enraged; and while he was enraged with the priests, the leprosy broke out on his forehead before the priests in the house of the LORD, beside the altar of incense. (2 Chronicles 26:19)

Melchizedek is thus a unique man with a unique designation as both king and priest! Melchizedek's specific identity does not detract from the powerful logic of the writer of Hebrews, who is appealing to his Hebrew Scriptures to substantiate his argument. He is saying in essence that even the Hebrew Scriptures recognize a priesthood that is distinct from that of Aaron, and that furthermore, this priesthood antedates Aaron's priesthood by literally hundreds of years.

John Calvin  points out that it is remarkable that Melchizedek lived with Sodom on one side and the Canaanites on the other, and yet he was a righteous king and priest. This shows that God can raise up a godly witness for Himself when and where He pleases. Here are Calvin's comments...

 It was doubtless no common thing that in a country abounding in the corruptions of so many superstitions, a man was found who preserved the pure worship of God; for on one side he was nigh to Sodom and Gomorrah, and on the other to the Canaanites, so that he was on every side encompassed by ungodly men. Besides, the whole world was so fallen into impiety, that it is very probable that God was nowhere faithfully worshipped except in the family of Abraham; for his father and his grandfather, who ought to have retained true religion, had long before degenerated into idolatry. It was therefore a memorable fact, that there was still a king who not only retained true religion, but also performed himself the office of a priest. And it was doubtless necessary that in him who was to be a type of the Son of God all things excellent should be found: and that Christ was shadowed forth by this type is evident from the Psalm referred to; for David did not say without reason, “Thou art a priest forever after the order Melchisedec;” no, but on the contrary, by these words a sublime mystery was recommended to the Church.

WHO MET ABRAHAM AS HE WAS RETURNING FROM THE SLAUGHTER OF THE KINGS AND BLESSED HIM: ho sunantêsas hupostrephonti (PAPMSD) apo tes kopes ton basileon kai eulogesas (AAPMSN) auton: (Genesis 16:14-16; Isaiah 41:2,3)

This description is taken from the Genesis account ....

Genesis 14:17 (Moses writes) Then after his (Abraham's) return from the defeat of Chedorlaomer and the kings who were with him, the king of Sodom went out to meet him at the valley of Shaveh (that is, the King's Valley).

The writer of Hebrews substitutes Melchizedek in place of the King of Sodom as the one who met Abraham at the Valley of Shaveh (which is probably the modern day Kidron Valley [or see here])

Hebrews 7 is an important chapter for in it the writer establishes the following points...

1). Melchizedek’s historic identity

2). Melchizedek’s precedence and superiority to Levitical priesthood

a). Abraham paid him tithes (so then did Levi)

b). Melchizedek blessed Abraham

3). The need for a replacement of the Law because it...

a). Could not make perfect

b). Jesus was from another tribe not specified in Mosaic Law

4). The advantages of the Melchizedekian order...

a). A better hope = access to God (draw near)

b) A guarantee of a better covenant

c). A permanent priesthood

d). Salvation forever... and a priest Who ever intercedes

5). A dramatic contrast between Jesus' priesthood and the Levitical priesthood

a) Jesus is holy, innocent, undefiled, separated, exalted

b). Jesus offered up one sacrifice of himself for all time.

MELCHIZEDEK
Various Interpretations
of His Identity

John Calvin

And it was doubtless necessary that in him who was to be a type of the Son of God (see discussion of Typology) all things excellent should be found: and that Christ was shadowed forth by this type is evident from the Ps 110

Steven Cole (his sermons are highly recommended)

Melchizedek is a type of Christ in the dignity of his person. Everything we know about Melchizedek comes from Genesis 14:18-20, Psalm 110:4, and Hebrews 7. The first text is historical, the second is prophetic, and the third is theological. (open the Pdf of the entire sermon Why You Need to Know About Melchizedek)

Alfred Edersheim

That Melchizedek was not Christ Himself is evident from the statement that he was “made like unto the Son of God” (or “likened unto” Him, Heb. vii. 3); while it equally appears from these words, and from the whole tenor of Scripture, that he was a type of Christ.

Matthew Henry

The most general opinion is that he was a Canaanite king, who reigned in Salem, and kept up religion and the worship of the true God; that he was raised to be a type of Christ (see discussion of Typology), and was honoured by Abraham as such.

Kent Hughes

In the writer’s opening statement he concisely states the significance of the historical Melchizedek as a type of the ultimate priesthood of Christ: Melchizedek was “without father or mother, without genealogy, without beginning of days or end of life, [and] like the Son of God he remains a priest forever” (v. 3). Some have inferred from these words that Melchizedek must have been an angel who took on human form for Abraham, or even a pre-incarnate appearance of Jesus Himself. But such interpretations are unnecessary, because the writer is simply using a rabbinical method of interpretation from silence. His point is that the Genesis account does not mention Melchizedek’s parents or genealogy or when he was born or died, thereby providing a fitting type of what would be fleshed out in the qualifications of Christ (see discussion of Typology). (Hughes, R. K. Hebrews: An Anchor for the Soul. Volume 1.  Crossway Books; Volume 2)

H A Ironside

There is no reason to think of Melchisedec as a mysterious person, possibly supernatural, or even as some have supposed a preincarnate appearance of our Lord Jesus Christ. If any ask, "Who is Melchisedec?" the only proper answer is "Melchisedec." He was not Shem the son of Noah, nor Job of the land of Uz, nor Cheops the builder of the great pyramid, as some have endeavored to prove. He was, as is distinctly stated, Melchisedec, King of Salem.

KJV Commentary

What is true of Melchizedek (the type) only because of silence is intrinsically true of Christ (the reality). Melchizedek is without parents only in that they are unknown. (Dobson, E G, Charles Feinberg, E Hindson, Woodrow Kroll, H L. Wilmington: KJV Bible Commentary: Nelson)

John MacArthur

But Melchizedek is described as made like the Son of God (7:3), not as being the Son of God. I believe that Melchizedek was a historical human being, whose priestly ministry typifies that of Christ (see discussion of Typology), a man whom God designed to use as a picture of Jesus Christ. But we cannot be sure of the details of his identity. (MacArthur, John: Hebrews. Moody Press )

William MacDonald

We should not conclude that Melchizedek had no parents, that he was never born, and that he never died. That is not the point. The thought is that as far as his priesthood was concerned, there is no record of these vital statistics because his ministry as priest was not dependent on them. (MacDonald, W & Farstad, A. Believer's Bible Commentary: Thomas Nelson)

J Vernon McGee

Melchizedek is a type of Christ. (McGee, J V: Thru the Bible Commentary:  Thomas Nelson)

Henry Morris...

The usual interpretation ... is that he (Melchizedek) was made into a type of Christ since as a "King of Righteousness" (meaning of Melchizedek) and "King of Peace" (meaning of Salem), he appears and leaves the record suddenly, with no mention of either ancestry or death. It seems better to take the words literally, in which case they could be applicable to Christ Himself, appearing here to Abram in a theophany. This would also solve the problem of how such a godly king and priest as Melchizedek could be ruling a city in such an ungodly land as Canaan and, why, if he did, Abram would have had no other contact with him. The fact that he was "like unto the Son of God" (Hebrews 7:3) accords with one of Christ's pre-incarnate appearances; at His human birth, he became the incarnate Son of God forever. Melchizedek was also said to be a man (Hebrews 7:4), but the same is true in the case of other theophanies, one of which was likewise manifested to Abram (Genesis 18:2,22; 19:1-24). (Morris, Henry: Defenders Study Bible. World Publishing)

Dwight Pentecost

The question is often raised about whether what is recorded in Genesis 14 is a theophany; that is, a preincarnate appearance of the eternal Son of God. While many say it is, the context of Genesis 14 seems to argue against it. Every verifiable theophany in the Old Testament fulfills the purpose of bringing a message from God to men. But that is not the case here. Further, the details of the account—giving names and places—argue against it. Melchizedek could hardly be called the “king of Salem” unless he exercised legal authority there over an extended period of time. When the writer says he was “made like the Son of God,” he seems to imply that only those things had been recorded that could be used later by the writer of the Hebrews to reveal truth concerning Christ’s priestly office. Thus, in the historical context, Melchizedek is an individual, universal, timeless, unique priest whose ministry resulted in spiritual and material benefits; and he is never known outside of that picture. In this, as the writer of the Hebrews will show, he represents a perpetual foreshadowing of the priestly order Christ will fill. (Pentecost, J. D., & Durham, K. Faith that Endures: A Practical Commentary on the Book of Hebrews. Grand Rapids, MI: Kregel Publications)

Charles Ryrie

Melchizedek is clearly a type of Christ (see discussion of Typology). Everything known about him from the OT is found in Ge14:17-20 and Ps. 110:4. He was a great king-priest, and it is to his order of priesthood that Christ belongs. (The Ryrie Study Bible: New American Standard Translation: 1995. Moody Publishers)

C I Scofield

Melchizedek, a type of Christ the King-Priest.

C H Spurgeon

WE will not enlarge upon the story of Melchisedec, nor discuss the question as to who he was. It is near enough for us to believe that he was one who worshipped God after the primitive fashion, a believer in God such as Job was in the land of Uz, one of the world’s grey fathers who had kept faithful to the Most High God. He combined in his own person the kingship and the priesthood; a conjunction by no means unusual in the first ages. Of this man we know very little; and it is partly because we know so little of him that he is all the better type of our Lord, of whom we may enquire, “Who shall declare his generation?” The very mystery which hangs about Melchisedec serves to set forth the mystery of the person of our divine Lord.

Ray Stedman

Though some commentators have viewed Melchizedek as a preincarnate appearance of Christ, the phrase like the Son of God seems to militate against that. “Melchizedek thus was the facsimile of which Christ is the reality” (Howley 1969:552). To a modern congregation, this passage should be presented as a vivid picture of the help which is available for believers today from our great high priest who can give us righteousness and peace from within if we “come to the throne of grace to receive mercy and find grace to help us in our time of need.” (Stedman, Ray: Hebrews IVP New Testament Commentary Series or Logos)

Will Varner

A more popular interpretation is that Melchizedek was Christ Himself in some preincarnate form. Thus, he would have been like the Old Testament “angel of the LORD” (e.g., Ge 16:7-11; Ex. 3:2; Jud. 13:3-21). Proponents of this view point to the language of Hebrews 7:3: “Without father, without mother, without descent, having neither beginning of day nor end of life.” There are some serious problems, however, with this idea. Six times the writer of Hebrews cited Psalm 110:4 when stating that Jesus is a priest “after the order of Melchizedek” (5:6, 10; 6:20; 7:11, 17, 21). If Jesus actually was Melchizedek, He would not be said to be “after the order of Melchizedek.” Furthermore, language of similarity, not identity, is used to describe the relationship between the two. Hebrews 7:3 states that Melchizedek was “made like unto the Son of God,” not that he actually was the Son of God. Finally, Hebrews 7:15 states that Jesus is a priest “after the similitude [likeness] of Melchizedek,” not that he actually was Melchizedek. These verses indicate that Melchizedek was an individual who was a type of Christ, not that he actually was the preincarnate Christ. (Israel My Glory : Volume 51 Issue 6. 1999)

John Walvoord

Although some have thought that Melchizedek was actually a theophany, that is, an appearance of Christ in the form of Melchizedek to Abraham, the more probable view is that Hebrews means only that Melchizedek, unlike Aaronic priests, had no recorded genealogy. He was a priest independent of his father or his successor. In other words, he was not dependent on his genealogy, in sharp contrast to the Aaronic priesthood which depended upon it completely. The predecessors and successors of Melchizedek are not mentioned in the Bible, and the validity of the Melchizedek priesthood does not rest upon this background.

Warren Wiersbe

Melchizedek was a man (see Hebrews 7:4), so he had to have had a mother and a father. But there is no record of his genealogy (“descent”) in the Old Testament; and this is significant because most great persons in the Old Testament have their ancestry identified. (Wiersbe, W: Bible Exposition Commentary. 1989. Victor)

Mormons (which I consider a non-Christian cult! [see critique of Mormonism] This note is only for completeness)

Claim that their male members are priests of the order of Melchizedek and that their prophet, Joseph Smith, held both the Aaronic and Melchizedek priesthoods. But this is a wholly gratuitous claim since it rests on no objective appointment by God but only on a subjective assertion in which they take this honor upon themselves.

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In "The Way into the Holiest" F B Meyer writes that...

History gives its unanimous judgment against the temporal and the spiritual power being vested in the same man. In Israel the two offices were kept rigorously separate; and when, on one occasion, a king passed the sacred barrier, and, snatching up a censer, strode into the inner court, he was at once followed by the remonstrances of the priestly band, whilst the white brand of leprosy wrote his doom upon his brow; "and he himself hastened to go out, because the Lord had smitten him." But the simple monarch of whom we write, living before gathering abuses forbade the union, combined in his person the royal scepter and the sacerdotal censer. And herein he foreshadowed the Christ.

Jesus is King and Priest. He is King because He is a priest. He is highly exalted, demanding homage from every knee, and confession from every lip, because He became obedient to the death of the cross. He bases His royal claims, not on hereditary descent, though the blood of David flowed in His veins; not on conquest or superior force; not on the legislation that underpins the kingdom of heaven among men: but on this, that He redeemed us to God by His blood. He is the King of glory, because He is the Lamb of God which taketh away the sin of the world. The cross was the stepping-stone to His throne.

And He cannot fulfill his office as Priest unless he be first recognized as King. Many fail to derive all the blessing offered to men through the Priesthood of Christ, because they are not willing to admit His claims as King. They do not reverence and obey Him. They do not open the whole of the inner realm to His scepter. They endeavor to serve two masters; and to stand well with empires as different as light and darkness, heaven and hell, God and Satan.

There must be consecration before there can be perfect faith; coronation before deliverance; the King before the Priest.

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