When did Kuradeel join Laughing Coffin?
Based off of the Chronological timeline, Laughing Coffin recruited Kuradeel after his dual with Kirito. Aside from Kuradeel wishing to kill Kirito, no other reason for joining Laughing Coffin is mentioned.

October, 2024
Although the guild was vanquished during the crusade, PoH, who avoided the crusade, watched Kirito and Kuradeel's duel on the 74th Floor. After hearing that Kuradeel wished to kill Kirito, PoH approached Kuradeel some time prior to Kirito's mandatory training session with Kuradeel and Godfree. PoH then invited Kuradeel to his guild and taught him how to make paralysis poison.