After the order of ...

And Melchizedek king of Salem brought forth bread and wine: and he was the priest of the most high God. (Genesis 14:18)

The LORD hath sworn, and will not repent, Thou art a priest for ever after the order of Melchizedek. (Psalms 110:4)

In Hebrews we hear of him again: As he saith also in another place, Thou art a priest for ever after the order of Melchisedec. (Hebrews 5:6)

In Hebrews he is mentioned several times:

Called of God an high priest after the order of Melchisedec. (Hebrews 5:10)

Whither the forerunner is for us entered, even Jesus, made an high priest for ever after the order of Melchisedec. (Hebrews 6:20)

Several verses in Chapter 7 describe this priest of the most high God – a servant of the Lord God before Abram became Abraham. A priest described as the Prince of Peace.

If therefore perfection were by the Levitical priesthood, (for under it the people received the law,) what further need was there that another priest should rise after the order of Melchisedec, and not be called after the order of Aaron? (Hebrews 7:11)

And it is yet far more evident: for that after the similitude of Melchisedec there ariseth another priest, Who is made, not after the law of a carnal commandment, but after the power of an endless life. For he testifieth, Thou art a priest for ever after the order of Melchisedec. (Hebrews 7:15-17)

(For those priests were made without an oath; but this with an oath by him that said unto him, The Lord sware and will not repent, Thou art a priest for ever after the order of Melchisedec:) By so much was Jesus made a surety of a better testament. (Hebrews 7:21-22)

There is a wealth of information on differing views of Melchizedek/Melchisedec and I will not delve into them. Hebrews explains quite well why a descendant of Judah rather than Levi, fulfills prophecy. Yet, a study of Genesis would be incomplete without these additional verses for context.