Melchisedec with the sons of Levi (the Levitical priesthood)
When Abraham recognized Melchisedec as the priest of the Most High God and gave him tithes in Genesis 14, it is accounted that the whole Levitical priesthood did likewise since Abraham is the Father of the Nation Israel and they being his seeds were at that time in his loins.
Levitical priesthood with Christ's NOW priesthood
By God's design the Levitical priesthood was to teach the other eleven tribes of Israel the Law of the Old Testament. For God's blessings, the Law demanded perfection (continual performance in well-doing). Part of their instruction declared that Israel was to be holy as He was holy (Leviticus 20:7-8), but under the Old Law there was no provision for perfection. Thus there was no perfection by that priesthood. BUT Jesus Christ, made by God to be an eternal, everlasting Great High
Priest after the similitude of Melchisedec's priesthood, can now perfect the people through the doctrine of the New Testament (i.e. a detailed explanation as to what God did for them through the blood and body of Christ with His continued
ministry as their Great High Priest) [perfection...Holiness...now available to Israel--Hebrews 10:10-14; 1 Peter1:15-16]
Christ's priesthood being after the order of Melchisedec
Hebrews 7:15-22 God, through the resurrection of His eternal Son, gave to Him honor and glory of a new priesthood that would last forever. His priesthood is similar to Melchisedec's priesthood in that it has some of the same characteristics; the most notable being that it is a continually abiding priesthood ordained by the MOST HIGH GOD.
From: A Dispensational Survey of the Epistles to the Hebrews
by Duane Gallentine