There are three characters of Christ's priesthood in the book of Hebrews: first, that of Aaron, because of our infirmities (Heb. 5:1-4); second, Phinehas, because Christ has made atonement to a holy God, therefore He has an everlasting priesthood (Heb. 7:27,8:3; Num. 25:10-13); and third, that of Melchizedek, king and priest, King of peace (Heb. 7:1-3).
It is through the Priest that we draw near unto God. This Priest, Christ, is after the order of Melchizedek and has no beginning nor end. His work for us is everlasting intercession.
He is Priest of the most high God-God's title as supreme over all, also His millennial name. In the millennial day Christ will be a Priest upon His throne. This is the earthly part.
For us He has an uninterrupted priesthood after the order of Melchizedek. As a Priest, He had no father or mother from whom He should receive His credentials.
Aaron was of the tribe of Levi. There had been no priesthood after the tribe of Judah, yet Christ came from Judah, not to pass His priesthood on to another, as did Aaron, but to be a Priest after the power of an endless life. Because there was a change of the priesthood, there must also be a change in the sacrifice and the covenant. The new covenant would be of the same character as the new order of priesthood.
Christ's priesthood was established by oath (Heb. 7:21); Aaron's was not. Such a High Priest became us, holy, harmless, and undefiled, separate from sinners, exalted above the heavens.
The priests under the law had the same infirmities as other men; Jesus (His eternal name) has a glorified priesthood according to the power of an endless life (Heb. 6:20).
The first covenant was made with Israel and the second must also be made with Israel. Christians are not under any covenant because they are a heavenly people and there are no covenants in heaven. However, they receive all of the blessings of that covenant now, which will not be made with Israel until the beginning of the millennial day (Ezek. 39:29; Jer. 31:31-34).