Yes - Catholic believe in Purgatory. Here are some of the biblical references.
A State After Death of Suffering and Forgiveness
Matt. 5:25,18:34; Luke 12:58-59 - these verses allude to a temporary state of purgation called a "prison." There is no exit until we are perfect, and the last penny is paid.
Matt. 5:48 - Jesus says, "be perfect, even as your heavenly Father is perfect." We are only made perfect through purification, and in Catholic teaching, this purification, if not completed on earth, is continued in a state we call purgatory.
Matt. 12:32 - Jesus clearly provides that there is forgiveness after death. Forgiveness is not necessary in heaven, and there is no forgiveness in hell. This proves that there is another state after death, and the Church for more than 2,000 years has called this state purgatory.
Luke 12:47-48 - when the Master comes (at the end of time), some will receive light or heavy beatings but will live. This state is not heaven or hell, because in heaven there are no beatings, and in hell we will will no longer live with the Master.
Luke 16:19-31 - in this story, we see that the dead rich man is suffering but still feels compassion for his brothers and wants to warn them of his place of suffering. But there is no suffering in heaven or compassion in hell. So where is the rich man? He is in purgatory.
1 Cor. 15:29-30 - Paul mentions people being baptized on behalf of the dead, to atone for their sins. These people cannot be in heaven because they are still with sin, but they also cannot be in hell because their sins can no longer be atoned for. They are in purgatory.
Phil. 2:10 - every knee bends to Jesus, in heaven, on earth, and "under the earth" which is the realm of the righteous dead, or purgatory.
2 Tim. 1:16-18 - Onesiphorus is dead but Paul asks for mercy on him. But there is no need for mercy in heaven, and there is no mercy given in hell. Where is Onesiphorus? He is in purgatory.
Heb. 12:14 - without holiness no one will see the Lord. We need final sanctification to attain true holiness before God, and this process occurs during our lives and, if not completed, in the state of purgatory.
Heb. 12:23 - the spirits of just men who died in godliness are "made" perfect. They do not necessarily arrive perfect. They are made perfect after their death. But those in heaven are already perfect, and those in hell can no longer be made perfect. These spirits were in purgatory.
1 Peter 3:19; 4:6 - Jesus preached to the spirits in the "prison." These are the righteous souls being purified for the beatific vision.
Rev. 21:4 - God shall wipe away their tears, and there will be no mourning or pain, but only after the coming of the new heaven and the passing away of the current heaven and earth. But there is no morning or pain in heaven, and God will not wipe away their tears in hell. These are the souls experiencing purgatory.
Rev. 21:27 - nothing unclean shall enter heaven. Even the propensity to sin is uncleanliness. It is amazing how many Protestants do not want to believe in purgatory. Purgatory exists because of the mercy of God. If there were no purgatory, this would also likely mean no salvation for most people. God is merciful indeed.
Gen. 50:10; Num. 20:29; Deut. 34:8 - here are some examples of ritual prayer and penitent mourning for the dead for specific periods of time. The Jewish understanding of these practices was that the prayers freed the souls from their painful state of purificatioin, and expedited their journey to God.
Baruch 3:4 - Baruch asks the Lord to hear the prayers of the dead of Israel. Prayers for the dead are unnecessary in heaven and unnecessary in hell. These dead are in purgatory.
Zech. 9:11 - God, through the blood of His covenant, will set those free from the waterless pit, a spiritual abode of suffering which the Church calls purgatory.
2 Macc. 12:43-45 - the prayers for the dead help free them from sin and help them to the reward of heaven. Those in heaven have no sin, and those in hell can no longer be freed from sin. They are in purgatory. Luther was particularly troubled with these verses because he rejected the age-old teaching of purgatory. As a result, he removed Maccabees from the canon of the Bible.