There are three characters of Christ's priesthood in the book of Hebrews: first, that of Aaron, because of our infirmities (Heb. 5:1-41For every high priest taken from among men is ordained for men in things pertaining to God, that he may offer both gifts and sacrifices for sins: 2Who can have compassion on the ignorant, and on them that are out of the way; for that he himself also is compassed with infirmity. 3And by reason hereof he ought, as for the people, so also for himself, to offer for sins. 4And no man taketh this honor unto himself, but he that is called of God, as was Aaron. (Hebrews 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. 25:10-1310And the Lord spake unto Moses, saying, 11Phinehas, the son of Eleazar, the son of Aaron the priest, hath turned my wrath away from the children of Israel, while he was zealous for my sake among them, that I consumed not the children of Israel in my jealousy. 12Wherefore say, Behold, I give unto him my covenant of peace: 13And he shall have it, and his seed after him, even the covenant of an everlasting priesthood; because he was zealous for his God, and made an atonement for the children of Israel. (Numbers 25:10‑13)); and third, that of Melchizedek, king and priest, King of peace (Heb. 7:1-31For this Melchisedec, king of Salem, priest of the most high God, who met Abraham returning from the slaughter of the kings, and blessed him; 2To whom also Abraham gave a tenth part of all; first being by interpretation King of righteousness, and after that also King of Salem, which is, King of peace; 3Without father, without mother, without descent, having neither beginning of days, nor end of life; but made like unto the Son of God; abideth a priest continually. (Hebrews 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.
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. 6:2020Whither the forerunner is for us entered, even Jesus, made an high priest for ever after the order of Melchisedec. (Hebrews 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. 39:2929Neither will I hide my face any more from them: for I have poured out my spirit upon the house of Israel, saith the Lord God. (Ezekiel 39:29); Jer. 31:31-3431Behold, the days come, saith the Lord, that I will make a new covenant with the house of Israel, and with the house of Judah: 32Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the Lord: 33But this shall be the covenant that I will make with the house of Israel; After those days, saith the Lord, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people. 34And they shall teach no more every man his neighbor, and every man his brother, saying, Know the Lord: for they shall all know me, from the least of them unto the greatest of them, saith the Lord: for I will forgive their iniquity, and I will remember their sin no more. (Jeremiah 31:31‑34)).