Although the word “purgatory” is not mentioned in the Bible, the reality of purgatory is clearly mentioned there. The words “Trinity” and “incarnation” and “Bible” are not mentioned in the Bible, but these realities are revealed there. Jesus speaks of sins to be forgiven in the life to come (Matthew 12:23). Saint Paul speaks of those saved in the next life “through fire” (1 Corinthians 3:15). Saint Peter speaks of the “spirits in prison” (1 Peter 3:18-20) and of the Gospel “preached to the dead” (1 Peter 4:6). The Bible explicitly tells us to pray for the dead (2 Maccabees 12:44-46). Besides being found in Sacred Tradition, which explicitly affirms the existence of purgatory, it is clearly that the doctrine of it existence is also found in Sacred Scripture.

2 Maccabees 12:43-46
43
He then took up a collection among all his soldiers, amounting to two thousand silver drachmas, which he sent to Jerusalem to provide for an expiatory sacrifice. In doing this he acted in a very excellent and noble way, inasmuch as he had the resurrection of the dead in view;
44
for if he were not expecting the fallen to rise again, it would have been useless and foolish to pray for them in death.
45
But if he did this with a view to the splendid reward that awaits those who had gone to rest in godliness, it was a holy and pious thought.
46
Thus he made atonement for the dead that they might be freed from this sin.

Matthew 12:32
32
And 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 in the age to come.

[b:2j7zc6zk]Would it be a sin for a Catholic not to believe in the existence of purgatory[/b:2j7zc6zk]?

Yes, it would be a mortal sin of heresy for such a Catholic. Read about purgatory in the Catechism of the Catholic Church (numbers 1030-1032). Read about heresy in the same book (number 2089).