If we get caught up only in the curiosity of this Melchizedek character in this chapter, we miss out on a lot. However, the little that Scripture speaks of him is central to understanding the argument the Hebrews writer is making. To get this chapter correctly, it’s wise to read Genesis 14 and Psalm 110 (although the relevant parts of both are quoted in Hebrews as well).
Chapter 6 ends with a quote from Psalm 110, one we’ve already seen in Hebrews prior to this, the promise that the Messiah will be a priest forever after the order of Melchizedek. Further, that v.4 from the Psalm affirms that this is a promise sworn by God, an oath (it’s going to happen, as sure as God keeping his promise to Abraham).
This is a long chapter so I’ll summarize where it’s going. Abraham tithed to high priest Melchizedek, a priest who existed before the priest system that was set up through the Law handed down through Moses. (How did he get the job? God gave it to him.) Melchizedek, in turn, blessed Abraham (and, by proxy, Levi, his future lineage). And since those who give blessings are greater than those receiving the blessings, we should understand that the priest in the order of Melchizedek are greater than the the priestly line that comes from Levi.
Psalm 110 promises that the future Messiah will be a priest in the order of Melchizedek, an indication even in that day that the Levite priests had a time limit. The system that was set up for and through them was not perfect (a better rendering is probably “complete”, meaning that the law and priests and sacrificial system had its purposes but was not capable of fully accounting for the sin in the world or showing the full identity that God has for his people).
So, the Jews should have been expecting, at some point, this Messiah figure who would be in the order of Melchizedek, one who is outside of and greater than the system administered by the Levite priests. And the Hebrew writer is making the case that Jesus is indeed this Messiah and High Priest. Now, the lineage of Jesus is the tribe of Judah (descendant of David), which means he certainly isn’t a Levite (something that would be quarrelsome to the Jews if there wasn’t this Melchizedek angle).
But Jesus qualifies to be High Priest, not because he is a descendant of Levi but because he was appointed by God (like Melchizedek) to fulfill the oath God made in Psalm 110 and the fact that Jesus has (and will continue to) lead a perfect life. And ultimately, this is better. In the old agreement between God and His people, the priests had to change all of the time because they died off. However, Jesus serves permanently, his intercession on our behalf never ends!
And, unlike the Levites, continuous sacrifice is not necessary because Jesus does not have his own sins to atone for. He was able to offer the perfect sacrifice (himself) by the perfect High Priest (himself) which does the deal forever.
Now, a few random notes. Melchizedek is a bit of a mysterious figure but the ESV kind of adds to this unnecessarily with its translation about him having no beginning or end. It’s probably more clearly rendered as: “No mention is made of his father or mother or genealogy, nor the beginning or end of his earthly life.” Basically, it’s not that he doesn’t have a beginning or end or a mom and dad, it’s that the Bible doesn’t mention it. This, coupled with the explanation of his name being king of righteousness, king of Jerusalem, king of peace, are all designed to help point to Jesus, who will resemble these same things in word and deed.
Also, we get an explanation of the fulfillment of the Law in Jesus where it is noted that there is a change in the law when there is a change in the priesthood. The tension here with the Law, though, is a bit difficult when the writer refers to it as “useless” and “weak”. I wonder if we are to think of the Law in this way specifically in relation to Jesus and what He accomplishes. As understood from Psalm 110, it was understood that the Levite priesthood was only temporary, which implied that it wasn’t capable of fulfilling the entirety of God’s purposes. Now that time has come, and any attempt to attribute the final, complete and perfect work that Jesus has done as if it could have been delivered from the Law are weak and useless.