Where does “priest, prophet and king” appear in the Bible?  Are we as good Catholics considered priest, prophet and king?

I don’t think you will find that phrase “priest, prophet, king” in the Bible, but you will in the Catechism (nos. 897-913).  But don’t worry, there are many things we do and believe that are not specifically stated in the Bible, such as Mass on Sunday and that God is a Trinity of persons.  However, all over the pages of the Old Testament you will find priests (Aaron and members of the Levite tribe), prophets (Elijah, Elisha, Moses), and kings (David, Solomon, etc.)  All of these Old Testament persons in some way pre-figure Jesus: the true priest, prophet, and king. And in one sentence (1 Kings 1:45) you will find the three terms (priest, prophet, king), but applied to three different people:  “Zadok the priest and Nathan the prophet anointed David as king.”

Jesus is priest because: a) he offered the sacrifice of himself to God in propitiation for our sins, and b) he is the bridge between man and God. Jesus is prophet because of the holiness of his words and teachings and his predictions of the future; finally, Jesus is king, because as God he has the fullness of power to carry out the duties of his kingly office: legislator, executor, and judge. And on his cross he was labeled the “King of the Jews” and when interrogated by Pontius Pilate he clearly admitted “Yes, I am a king.”
 
By our Baptism we are grafted onto and into Jesus Christ, and are made partakers in the divine nature.  By virtue of our baptism, all members of the faithful share in the triple mission of Christ: priest, prophet, and king.  The Code of Canon Law and other documents of the Church describe these roles as the Mission of Sanctifying (Sacraments), the Mission of Teaching, and the Mission of Ruling.  These offices are part and parcel of the ordained ministry: sacraments, preaching and governance.   But lay people too – as taught by the Second Vatican Council in “Lumen Gentium” 31 and later codified in the Code of Canon Law – also participate in the priestly, prophetic, and kingly mission of Christ.  They do so in various ways. 
 
Their priestly role (to sanctify) is normally carried out through their generous spirit of sacrifice in their daily work and in their family; but in a ritual way lay people also participate and cooperate in the Sacramental (priestly/sanctifying) mission of the Church as acolytes, lectors, extraordinary ministers of Holy Communion, and such. 
 
Their prophetic role (to teach) is normally carried out in the workplace, in their community, and in the family by their example and also by their friendly and mind-expanding conversations with all they come in contact with.  In a more organized and institutional way, lay people carry out their prophetic dimension by teaching CCD, or serving as a FOCUS (Fellowship of Catholic University Students) missionary on college campus for a few years.
 
Finally, their kingly role (to rule) is normally carried out at home as parents, but they can also cooperate with the local bishop and priests in the governance of the Church by serving as the chancellor of the diocese, or as a judge on a marriage tribunal, or on the finance council of the local parish.