First of all, the Bible does not contain the full revelation of God. This full revelation is found in both Scripture (the Bible) and Tradition (teachings of the Catholic Church which Jesus established). It is therefore not necessary that the Bible contain every Christian doctrine. For example, there is no mention in the Bible that the Holy Spirit is God or that God is a Trinity.

The doctrine of Purgatory is not directly mentioned in the Bible either, but it is indirectly. 2 Machabees 12:46 says that "it is therefore a holy and wholesome thought to pray for the dead, that they may be loosed from sins." Jesus said in Matthew 12:32 that "whoever speaks a word against the Son of Man will be forgiven; but whoever speaks against the Holy Spirit will not be forgiven, either in this age or the age to come" which implies that expiation can occur after death. St. Paul said that after death everyone's works will be judged and if the test is failed, "that one will suffer loss; the person will be saved, but only as through fire" (1 Corinthians 3:15). This loss cannot refer to Hell, since people cannot be saved there, but can only refer to Purgatory.