Cloud appears in four games in this series, Kingdom Hearts, Kingdom Hearts II, Kingdom Hearts: Chain of Memories, and Kingdom Hearts Re:coded. In the latter two, he appears in the Olympus Coliseum, where he is either a boss (Chain of Memories) or helps out the player in some other way (Re:coded).
In Kingdom Hearts, Cloud has been hired to kill Hercules by Hades. Before he can fight him, though, he has to fight Sora in the Olympus Coliseum. After the fight is over, Hades sics Cerberus on the group, with Hercules saving them. With all of the events ending, Cloud mentions to Sora that he is searching for someone, although this is never revealed.
It's in Kingdom Hearts II that Cloud makes another appearance, with him searching for Sephiroth but also being searched for by Tifa Lockhart. In the game, he wears the clothes he wore in the Advent Children film. He helps out Leon and his team when the Heartless invade Hollow Bastion. If the fight with Sephiroth is played out and won, he appears again and disappears while fighting Sephiroth.
