How is Christ our great high priest?

Sam Waldron explains in this video that the author of Hebrews is arguing that a new and greater priesthood has arisen. He wants his Jewish readers to understand that this new priesthood is something that is predicted in the Old Testament. 

Specifically, it is discussed in Psalm 110:4 (NKJV) – “The Lord has sworn
 and will not relent,
‘You are a priest forever
 according to the order of Melchizedek.’” Even though Jesus was not a Levitical priest, He was a greater priest.

Hebrews 4:14-16 (NKJV) - "Seeing then that we have a great High Priest who has passed through the heavens, Jesus the Son of God, let us hold fast our confession. For we do not have a High Priest who cannot sympathize with our weaknesses, but was in all points tempted as we are, yet without sin. Let us therefore come boldly to the throne of grace, that we may obtain mercy and find grace to help in time of need."

Now, when Hebrew 717 speaks of Christ as the order of Melchizedek, there's a great deal going on there. Fundamentally, that language is picked up because the author is arguing the point that a new and greater priesthood has arisen. And he wants to make sure for his readers' sake, his Jewish readers' sake, that they understand that this rising of a new and greater priesthood is something predicted in the Old Testament. So the citation of Psalm 110 about Christ being a priest after the order of Melchizedek is crucial to his argument because it shows that the Aaronic priesthood, the Levitical priesthood, was not the last final word in terms of priesthoods in the Bible. That a new priesthood, the priesthood after the order of Melchizedek, was predicted.

And that even though Jesus was not a Levitical priest, He could still be a priest, though not a Levitical priest, a greater priest, because He is a priest after the order of Melchizedek.