The Aaronic priesthood is characterized by atonement and intercession: that of Melchizedek is characterized by power and Messing. Christ is "the high priest of our profession." as Christians. He will be, as a result, "priest of the Most High God" God's millennial name.
The order of His priesthood is that of Melchizedek. Its exercise at present is after the pattern or character of Aaron, that is, intercessional. He was "called" to the priesthood by the word of Him that said unto Him, "Thou art My Son, to-day have I begotten Thee." Heb. 5:5. This has reference to His being the Son of God. as horn of a woman, and born in time on earth. Compare Psa. 2:7 and Luke 1:35. This is distinct from His being God's eternal Son.
He is installed in His priesthood after the order of Melchizedek as having gone on high when He had been rejected on earth, had died and risen, and had ascended to heaven. Compare Heb. 5:6 and Psa. 1
He was perfected for His priesthood (especially for its present exercise), "in the days of His flesh." through strong crying and tears, and His pathway of sorrow and suffering. Then He went on high. (Heb. 5:7-9: Mar. 14:33-40: Luke 22:40-53.) Having gone through all this, He was "called of God a high priest after the order of Melchizedek" ( Heb. 5:10), when He ascended into the heavens. There and then He first practically exercised His priesthood. When He comes forth again He will exercise it after its true order, as Melchizedek. F.G. Patterson