Greetings in that Divine and Most Precious Name of Our Lord and Savior Jesus Christ!
Read this text in its context, it was written for Jewish converts to the Early Church, and so it speaks of the flaws of legalism. St Paul first describes this apparently contradicting story of Abraham and Melketsedek, king of Salem. This story was fundamental to the Early Church, as the story of Melketsedek prefigures the Divine Liturgy, as he offered a prayer service of thanksgiving with Abraham and his entourage after that battle, using bread and wine. So St Paul was drawing several meanings from this one piece of symbolic imagery, that of good father Abraham recognizing the priestly authority of Melketsedek, so much so as to offer a tithe of the spoils of the battle, even though from a strictly Mosaic legal perspective, Abraham was the only one entitled to receive tithes, being the ancestor of the priestly Levi. Jewish theology at the time would have ascribed Abraham's material success as being of the virtue of the Law and the Levitical priesthood, that is to say, St Paul was saying to a Jewish audience that while Abraham had indeed succeeded in a battle for material game precisely because of the universal authority of the Levitical priesthood (of which Abraham was a blood representative in its proto-state) even blessed Abraham acknowledged tithing to a different priesthood. This would have been puzzling to Jews at the time. Further, St Paul was intending to silence Jewish legalists in the Early Church who would have been arguing that bloodline Levites should be the rightful priests within the Church. St Paul was providing a historical and venerable instance where even Abraham (that is Levi) gave tithe and authority to another priesthood, under Melketsedek. Any good Pharisee would have been rightfully outraged, as the Jewish priesthood based its entire mandate and authority on the ethnicism of the Levitical covenant. This is why they would in the Church continue to insist on their authority. Many Christians might have been persuaded as such, even today in a modern era we struggle with how to understand how to co-opt the Law into the Church. St Paul's point then was to defend the legitimacy and authority of the Orthodox Church priesthood OVER the Jewish Levitical priesthood. Surely Levites could also become Orthodox priests, but their Levite ancestry did not give them preferential spiritual authority in the eyes of God, and as St Paul explained elsewhere that the Scriptures and the people attest that Abraham was a man who "pleased God" and so his example in itself is authoritative on this matter. If father Abraham could recognized the priesthood of Melketsedek, than surely even a Pharisee convert could recognize the spiritual and ecumenical authority of the Orthodox Church.
Now in regards to Orthodox human priests and Jesus Christ's priesthood, St Paul is also arguing for multiple priesthoods in a sense. St Paul did not necessarily inviolate Abraham's own priesthood in his explanation of the story with Melketsedek, rather St Paul seems to affirm the priesthoods of both men simultaneously. So it can be inferred also than that St Paul's first point about the "Melketsedek Priesthood" was to explain the spiritual legitimacy and authority of Christian priests over Jewish Levites who were contemporaries.
The second point of the latter half explains Christ's own priesthood. Jesus Christ is the true Chief Priest, above the Levites, as His Priesthood never ends, and He has no sins of His own to account for. Jesus Christ as a Priest eternally petitions on the behalf of humanity, but it is deeper than this, because that is the human aspect of Jesus Christ, but the Divine Word not only speaks on behalf of humanity but acts directly for their salvation. Not only does Jesus Christ act as a priest by praying on our behalf, but He is the very source of the answers of those same said prayers!
How are Orthodox Priests still priests if Jesus Christ is the Chief Priest? They are lesser priests so to speak, who serve within the Ministry of Jesus Christ in the Apostolic Church. With Jesus Christ as the True Patriarch of our Church, the True High Priest, all those other priests serve and minister under Him, just as earthly priests serve under the Bishop. The Bishop does not necessarily lord over the lesser priests in a spiritual sense, rather the Bishop represents a seat of spiritual authority from where it stems forth. If Jesus Christ is the True High Priest sitting at the right hand of the Father ministering eternally on our behalves, then these lesser Orthodox priests we have here on Earth are ministering under Jesus Christ's central spiritual authority. The Priesthood of Jesus Christ stems out from Him, He is the Source, and our priests which we have here receive His authority. With the Levites the authority of their priesthood came from themselves, of themselves, by the virtue or privilege of their ancestry and genealogical background, where as the priesthood of the Orthodox Church and our clergy's authority is by no means of themselves, it comes directly from Jesus Christ.