He makes his only brief appearance in Genesis 14:18-20. We know that he was the king of Salem, and that he was a priest. He had the authority to make an offering to God and to bless Abram. Abram recognized his authority and tithed to him.
Some say that the offering of Melchizedek was a sort of pre-figuring of a Korban Todah (sacrifice of thanksgiving/praise), because of the bread and wine involved (though I think that Korban Todah also involved a lamb typically... I'll double check that and get back with you).
There are also the interesting words in Psalm 110:4 "The Lord has sworn and will not change his mind, 'You are a priest forever after the order of Melchizedek.'" Those words would have originally applied to Solomon, but we also know that they are a prophecy of Christ (see Hebrews 7).
Christ is a king, the King of Kings. He is also the definitive High Priest, not according to the Mosaic law (He was from the tribe of Judah, not a Levite), but according to the order of Melchizedek. I would say that He is the bringer of a new law that fulfills the old, ineffective one.
I suppose this doesn't really have any answers in it. I'm just thinking "out loud" so to speak.