Hebrews 5:5-6 meaning

The previous verse indicated that priests must be called by God to have a legitimate claim to priestly duties. The Pauline Author quotes from Psalm 2:7, 110 to show that Christ did not glorify Himself so as to become a high priest (v 5), but that He was called by God to be a priest. He who said to Him, "You are My Son, today I have begotten You (v 5)." God made Jesus an heir who will one day rule over all, and also appointed Him as high priest. The Pauline Author points out that God says also in another passage, "You are a priest forever according to the order of Melchizedek" (v 6). Jesus's priestly appointment will never end. So Christ meets this qualification for priesthood since He did not appoint Himself but was appointed by God.

The title Son and the role of priest were bestowed upon Jesus after He finished serving His father faithfully and obediently to His death. Just as being human was a prerequisite to being appointed High Priest among men, although Jesus was already the Son of God from the beginning, He received the reward of the title Son for His faithfulness as a human. 

The Pauline author will explain in Chapter 7 what he means by saying that Jesus is a priest according to the order of Melchizedek. He is quoting Psalm 110:4, where God is speaking to His Anointed One (Messiah). Drawing a comparison between Jesus and Melchizedek is meant to show that Jesus is a greater priest than the Levites (the Jewish priesthood), because Melchizedek was a greater priest than the Levites (Genesis 14:18-20). Melchizedek predated the Levites and was paid a tithe by Abraham, father of the nation of Israel. Melchizedek's name means "king" and "righteousness." He was the King of Salem, which means "peace," so his name means "king of righteousness and king of peace," representative of Christ. 

The point is that Melchizedek is a representation of our high priest, Christ: a king-priest of righteousness and of peace. Christ remains our high priest forever.