Kingdom of Heaven doesn't take place during a specific Crusade, per se, but rather depicts the events that preceded one. The movie focuses on the events right before the Third Crusade (although it does somewhat condense time, events moved much slower than the movie shows but that is to be expected), specifically the end of the reign of King Baldwin IV 'the leper' (1174-85) and Saladin's Siege of Jerusalem (1187). The very end of the movie includes Balian joining Richard I for the Third Crusade (1189-1192), but none of the events from that crusade are shown in the movie's story.

Edit: I should emphasise that my account above is the actual historical dates. The movie's plot is supposed to begin in 1184, so at the very end of Baldwin IV's reign. Baldwin's health was in serious decline after 1183, and he had his nephew Baldwin V crowned co-king near the end of that year. He also attempted to have Sybilla and Guy's marriage annulled, a clear sign of his displeasure with the couple, but failed. In the movie Baldwin is still at least mostly functional during the beginning of Balian's time in the Holy Land, not nearly as unwell as he would have been had Balian actually arrived in 1184/5, with his sickness progressing along with the plot. For this reason I think the movie could be considered to reasonably be representing a time period earlier than 1184, even if that is its stated starting year.