Genesis was written with the assumption that the readers already know who Melchizedek was (or is).

After his return from the defeat of Chedorlaomer and the kings who were with him, ... Melchizedek king of Salem brought out bread and wine. (He was priest of God Most High.) -- Genesis 14:17 (ESV)

Unfortunately this history of Melchizedek is lost.

However, Psalm 110 states:

The LORD has sworn and will not change his mind,

“You are a priest forever after the order of Melchizedek.”

-- Psalm 110:4 (ESV)

And since Jesus quoted Psalm 110 with the assumption that the Pharisees agreed that Psalm 110 was a prophecy about the coming Christ, it implies Jesus is the "priest" of which David prophesied.

Now while the Pharisees were gathered together,

Jesus asked them a question, saying, “What do you think about the Christ? Whose son is he?” They said to him, “The son of David.” He said to them, “How is it then that David, in the Spirit, calls him Lord, saying,

‘The LORD (YHVH) said to my Lord, sit at My right hand, until I put your enemies under your feet’?

If then David calls him Lord, how is he his son?”

And no one was able to answer him a word, nor from that day did anyone dare to ask him any more questions.

-- Matthew 22:41-46 (ESV); Mark 12:35–37; Luke 20:40–44

This still does not necessary mean Jesus is Melchizedek himself. Only that he was considered a priest after the order of Melchizedek like Levi and his sons and the generations after them were priests of the same order.

We have this as a sure and steadfast anchor of the soul, a hope that enters into the inner place behind the curtain, where Jesus has gone as a forerunner on our behalf, having become a high priest forever after the order of Melchizedek.

For this Melchizedek,

• king of Salem,
• priest of the Most High God,
• met Abraham returning from the slaughter of the kings and blessed him, and to him Abraham apportioned a tenth part of everything.
• He is first, by translation of his name, king of righteousness, and
• then he is also king of Salem, that is, king of peace.
• He is without father or mother tor genealogy, having neither beginning of days nor end of life, but resembling the Son of God he continues a priest forever.

See how great this man was to whom Abraham the patriarch gave a tenth of the spoils! And those descendants of Levi who receive the priestly office have a commandment in the law to take tithes from the people, that is, from their brothers, though these also are descended from Abraham. But this man who does not have his descent from them received tithes from Abraham and blessed him who had the promises. It is beyond dispute that the inferior is blessed by the superior. In the one case tithes are received by mortal men, but in the other case, by one of whom it is testified that he lives. One might even say that Levi himself, who receives tithes, paid tithes through Abraham, for he was still in the loins of his ancestor when Melchizedek met him.

Now if perfection had been attainable through the Levitical priesthood (for under it the people received the law), what further need would there have been for another priest to arise after the order of Melchizedek, rather than one named after the order of Aaron? For when there is a change in the priesthood, there is necessarily a change in the law as well. For the one of whom these things are spoken belonged to another tribe, from which no one has ever served at the altar. For it is evident that our Lord was descended from Judah, and in connection with that tribe Moses said nothing about priests.

This becomes even more evident when another priest arises in the likeness of Melchizedek, who has become a priest, not on the basis of a legal requirement concerning bodily descent, but by the power of an indestructible life. For it is witnessed of him,

“You are a priest forever, after the order of Melchizedek.” (Psalm 110:4)

For on the one hand, a former commandment is set aside because of its weakness and uselessness (for the law made nothing perfect); but on the other hand, fa better hope is introduced, through which we draw near to God.

And it was not without an oath. For those who formerly became priests were made such without an oath, but this one was made a priest with an oath by the one who said to him:

“The Lord has sworn and will not change his mind, ‘You are a priest forever.’” (Psalm 110:4)

This makes Jesus the guarantor of a better covenant.

The former priests were many in number, because they were prevented by death from continuing in office, but he holds his priesthood permanently, because he continues forever. Consequently, he is able to save to the uttermost those who draw near to God through him, since he always lives to make intercession for them.

For it was indeed fitting that we should have such a high priest, holy, innocent, unstained, separated from sinners, and exalted above the heavens. He has no need, like those high priests, to offer sacrifices daily, first for his own sins and then for those of the people, since he did this once for all when he offered up himself. For the law appoints men sin their weakness as high priests, but the word of the oath, which came later than the law, appoints a Son who has been made perfect forever.

-- Hebrews 6:19-7:28 (ESV)

If this information is correct we can conclude that Melchizedek was:

• The king of Salem
• A priest of God Most High
• He meet Abraham during his lifetime and received 10% of the spoils from him
• The Christ is a priest after the order of Melchizedek
• Melchizedek had no parents (unless the author meant they were unknown or unimportant)
• Melchizedek was made like the Son of God (either confirming that he was a god-like being or that he was a very righteous man)
• Melchizedek is an eternal priest (possibly also confirming that he was a god-like being or that the order that he established will continue forever)
• Melchizedek lives forever (possibly also confirming that he was a god-like being or that he is in heaven where he will live forever)

and

• The name Melchizedek means "king of righteousness"
• The name King of Salem means "king of peace"

What the author of Hebrews most likely meant is that Christ replaced the human priesthood system that was originally established by Melchizedek during Abraham's lifetime before Jesus was born. Since the Jews in Jesus' lifetime were already used to a priest interceding for them, it would make sense to explain to them that Jesus had come to fulfill that role as a high priest. In Paul's letters to the gentiles, who are not used to Jewish priesthoods, he explains that Jesus is interceding or mediating for us which is essentially the same role but described differently to a different audience:

Christ Jesus is the one who died (more than that, who was raised) who is at the right hand of God, who indeed is interceding for us. -- Romans 8:34 (ESV)

For there is one God, and there is one mediator between God and men, the man Christ Jesus, who gave himself as a ransom for all, which is the testimony given at the proper time. -- 1 Timothy 2:5-6 (ESV)