Roman Catholic and Orthodox Christians believe in a priesthood, i.e. office of the ministers of religion, who have been commissioned ordained with the Holy orders of the Orthodox or Catholic church; but what is the biblical basis for this type of a separate priesthood from the priesthood of all believers?
First, its important to define the ministerial priesthood (as separate from the priesthood of all believers)
The ministerial or hierarchical priesthood of bishops and priests, and the common priesthood of all the faithful participate, "each in its own proper way, in the one priesthood of Christ." While being "ordered one to another," they differ essentially. In what sense? While the common priesthood of the faithful is exercised by the unfolding of baptismal grace --a life of faith, hope, and charity, a life according to the Spirit--, the ministerial priesthood is at the service of the common priesthood. It is directed at the unfolding of the baptismal grace of all Christians. The ministerial priesthood is a means by which Christ unceasingly builds up and leads his Church. For this reason it is transmitted by its own sacrament, the sacrament of Holy Orders.
There clearly are priests all over the Old Testament, Priests and Levites and Firstborn sons all set aside as holy for one reason or another.
And then you can see all the evidence in the New Testament of the sacramental being conferred from Apostles (of whom the bishops are their successors) to other ministers, who are the priests of Christ's Church.
All talk about the laying on of hands and the commissioning of special ministers to do priestly things.
Also, as Orthodox and Catholics recognize each other's priesthood as valid and our New Testament understand of the sacrament and succession is identical, so I don't believe this answer should matter which perspective you're coming from.