Jesus as Priest
Today's passage starts to move into a slightly different area. Hebrews started out by showing how Jesus was the superior Revelation and Son, and in this passage the writer starts to show us how Jesus is superior to the previous priests. He speaks a lot about Melchizedek (Genesis 14:17-24) here and again the thinking is a little complex and more Jewish in it's reasoning than our Western minds are used to. Here's an extended quote to help explain:
"The author begins to describe the priesthood of Jesus by illustrating how Jesus was a priest in the same way that Melchizedek was a priest. Having already mentioned Melchizedek in passing (5:6, 10; 6:20), the author now returns to discuss him at length. In this discussion, the author shows that the Levitical priesthood, very familiar to Jewish readers, has been replaced by a new order of priests who were foreshadowed and characterized by Melchizedek.
Melchizedek was king of the city of Salem and also a priest of God Most High. This description of Melchizedek comes from Genesis 14:18–20. He seems to have been an extraordinary man who served his people in both the offices of priest and king. While this was not that unusual in ancient times, it was unusual among God’s people. Not even King David served in both roles. In fact, God punished other kings who attempted to do both. When Saul offered sacrifices, God rejected him as king (1 Samuel 13). When Uzziah offered incense at the temple, God punished him (2 Chronicles 26). “Salem” may later have become the city of Jerusalem. Melchizedek, however, served as the king and priest. The appellation “God Most High” means that Melchizedek worshiped the one true God. Genesis 14:18–20 uses the term “God Most High” three times in three verses. (For more uses of the name, see Psalm 57:2; 78:35, 56; Daniel 3:26; 4:2, 24–25, 34; 5:18, 21; 7:18, 25; Mark 5:7; Luke 8:28; Acts 16:17.)
This passage refers to the time when Abraham was returning home after winning a great battle against many kings. Four kings in Abraham’s region had united and had conquered Sodom and other neighboring cities (Genesis 14:1–11). Abraham’s nephew Lot and his family lived in Sodom. When Abraham heard that Lot and his family had been captured, Abraham mobilized 318 men for battle. With a surprise attack, Abraham and his tiny band of men liberated Lot and the others who had been captured (Genesis 14:12–16).
After defeating the four kings, Abraham became the greatest power in the land, and Melchizedek met him and blessed him. Then Abraham took a tenth of all he had won in the battle and gave it to Melchizedek because Melchizedek was a priest of God Most High. By giving the tithe to Melchizedek, Abraham was giving the gift to God’s representative. Although these two men were strangers to each other, they shared a most important characteristic: Both worshiped and served the one God who made heaven and earth. This was a great moment of triumph for Abraham. He had just defeated an army and had freed a large group of captives. If he had any doubt in his mind about who had gained the victory, Melchizedek set the record straight by reminding Abraham, “Blessed be God Most High, who delivered your enemies into your hand” (Genesis 14:20 NIV). Abraham recognized that he and this man worshiped the same God.
The original readers of Hebrews would have known that Melchizedek was greater than Abraham because he was able to receive tithes and give a blessing (see 7:7). This argument may not carry the same logical forcefulness for readers today as it did then, but these early Jewish believers understood the argument...
...To bolster the argument, the author of Hebrews used what is not said in Genesis 14 as much or more as what is said. The Bible does not provide a genealogy for Melchizedek nor a record of his death. While the Bible does not supply details of Melchizedek’s life, most likely Melchizedek was a human king and priest who really did have parents, and thus was born and eventually died. Jewish theology and typology, however, is built only on what the Bible text says. Because in the Bible text there is no record of his father or mother or any of his ancestors, it is as though he didn’t have any. Because the text records no beginning or end to his life, it is as though Melchizedek never was born or died.
While some have taken this to mean that Melchizedek was an angel or a preincarnate appearance of Christ, the text does not support this. Rather, the contrast is being made between Melchizedek and Aaron’s priestly line, which depended entirely on genealogy. Priests in Aaron’s family succeeded upon the death of the prior priest, making the date of death extremely important. None of the apparatus of the Aaronic priesthood (Exodus 39) applied to Melchizedek, except God’s appointment. In this way, Melchizedek foreshadows Jesus, God’s special emissary.
Melchizedek did not become a priest because his father handed down the priesthood to him. The contrast with Aaron’s priesthood is clear. The lineage of priests was very important. Priests who could not prove their lineage were excluded from service (see Nehemiah 7:64).
With no record of beginning or end, Melchizedek remains a priest forever, resembling the Son of God (see also Psalm 110:4). Hebrews doesn’t say that Jesus resembled Melchizedek, but that Melchizedek resembled Jesus. Melchizedek was a real man, a servant of God, whose history is recorded in the book of Genesis in such a way as to make him resemble the one who would come and fulfill completely the offices of priest and king, and who would truly be “a priest forever.”" 
As I said, the reasoning isn't as "western" as we would like, but do you follow what he's doing here? If not, comment or question below!