Steve Gregg goes through the different views on who Melchizedek was, but shows pretty clearly that he was indeed Jesus, our high priest forever, replacing the Aaronic Priesthood. As verse 12 says, when there is a change in the priesthood, there must be a change in the law. The Old Covenant and Mosaic Law were over and done with for Jewish and Gentile Christians alike.
New King James Version (NKJV)
The King of Righteousness
7 For this Melchizedek, king of Salem, priest of the Most High God, who met Abraham returning from the slaughter of the kings and blessed him, 2 to whom also Abraham gave a tenth part of all, first being translated “king of righteousness,” and then also king of Salem, meaning “king of peace,” 3 without father, without mother, without genealogy, having neither beginning of days nor end of life, but made like the Son of God, remains a priest continually.
Need for a New Priesthood
11 Therefore, if perfection were through the Levitical priesthood (for under it the people received the law), what further need was there that another priest should rise according to the order of Melchizedek, and not be called according to the order of Aaron? 12 For the priesthood being changed, of necessity there is also a change of the law. 13 For He of whom these things are spoken belongs to another tribe, from which no man has officiated at the altar.
14 For it is evident that our Lord arose from Judah, of which tribe Moses spoke nothing concerning priesthood.[a] 15 And it is yet far more evident if, in the likeness of Melchizedek, there arises another priest 16 who has come, not according to the law of a fleshly commandment, but according to the power of an endless life. 17 For He testifies:[b]
“You are a priest forever
According to the order of Melchizedek.”[c]
18 For on the one hand there is an annulling of the former commandment because of its weakness and unprofitableness, 19 for the law made nothing perfect; on the other hand, there is the bringing in of a better hope, through which we draw near to God.
Greatness of the New Priest
20 And inasmuch as He was not made priest without an oath 21 (for they have become priests without an oath, but He with an oath by Him who said to Him:
22 by so much more Jesus has become a surety of a better covenant.23 Also there were many priests, because they were prevented by death from continuing. 24 But He, because He continues forever, has an unchangeable priesthood. 25 Therefore He is also able to save to the uttermost those who come to God through Him, since He always lives to make intercession for them.
26 For such a High Priest was fitting for us, who is holy, harmless, undefiled, separate from sinners, and has become higher than the heavens; 27 who does not need daily, as those high priests, to offer up sacrifices, first for His own sins and then for the people’s, for this He did once for all when He offered up Himself. 28 For the law appoints as high priests men who have weakness, but the word of the oath, which came after the law, appoints the Son who has been perfected forever.