I think first you have to establish evidence for a ministerial priesthood. The Old Testament Jews had a concept of “priesthood of believers” (Exodus 19:60) but they also had a ministerial priesthood that alone was authorized to offer sacrifices to God. So in the OT you had the High Priest, the ministerial priests under him and the priesthood of believers, the laity. We have a perfect parallel on the New Testament with Jesus as the High Priest, a ministerial (sacramental) priesthood and priesthood of believers (laity).
During His public ministry, Jesus gave His disciples (only the twelve, not all who followed Him) the authority and power to do the same things He was doing in His public ministry (healing the sick and casting out evil spirits) Matthew 10:1 (Matthew 9 shows Jesus, just prior to this, healing and casting out evil spirits). In Matthew 26:26-29, Jesus uses the language of sacrifice with the apostles when He first offers His body and then His blood – in Old Testament sacrifices the blood was drained (separated) from the sacrificial animal. So Jesus is here offering His body and blood in sacrifice for sin. And He asks his apostles, and only them, to perpetuate the offering of this sacrifice (“Do this in remembrance of me” Luke 22:19) In John 20:19-23 Jesus gives only His apostles the authority to forgive and retain sins, another aspect of His earthly ministry. And right before His Ascencion into heaven, He gives the Great Commission to His apostles to preach, teach and baptize, making disciples (Matthew 28:18-20), giving them His own authority. Not all Christians share that same authority.
One of the arguments Protestants have against the priesthood is that it is no longer necessary because there is no longer any sacrifice now that Jesus has died. But looking at Hebrews 9;23 indicates otherwise – it speaks of “better sacrifices” (plural, not singular) of the new covenant compared to the sacrifices of the old covenant. The only way the sacrifices of the new covenant can be better is that they actually take away sin, so how do we reconcile that with the idea of their being only one sacrifice that eliminates all other sacrifice and the need for a ministerial priesthood? Because the new sacrifices are Jesus Christ’s very own Body and Blood, Soul and Divinity offered up in an unbloody manner (not all sacrifice requires killing), which alone can atone for sin. Hebrews 13:10 mentions “we have an altar from which those who serve the tent have no right to eat.” “We” are Christians, and we have an “altar” and altars are for sacrifice, and something edible (“have no right to eat”) is offered upon it. It’s not a sacrifice of praise or worship like some Protestants try to interpret it because you can’t eat praise and worship. That something edible is Jesus, and He is offered up by priests acting in His person.
Jude 8-11, speaking of men who “reject authority” who “perish in Korah’s rebellion.” If you go back to the Old Testament, Korah’s rebellion was against the ministerial priesthood. There had to have been a ministerial priesthood established in the Church for the men Jude is speaking of to be guilty of “Korah’s rebellion.”
James 5, cited in another post ties in with all this because it speaks of calling the “elders” so that the person may receiving healing and have sins forgiven. An “elder” is a priest, not just another layman, otherwise why specify “elders” for prayer for healing and forgiveness?