One of the three Silver Haired Men [unofficial name], Kadaj is the leader of the group. He is evil, cruel and manipulating, showing no remorse for his actions and not letting anyone get in the way of his goal.
Along with Loz and Yazoo, Kadaj is searching for 'Mother' [Jenova]. However little does he realize that he is actually doing Sephiroth's bidding, through the desire that Sephiroth has to return.
Although Sephiroth is the ultimate enemy in Advent Children, Kadaj is the main enemy. Kadaj's plan is to use the children to bring about the Reunion. Enticing them to do his bidding by offering them a new existence, and telling them of the planet making the children suffer by affecting them with Geostigma.
Kadaj's ultimate showdown is with Cloud, the enemy of Sephiroth, also being the enemy of the three men.
Character note: the three SHM all represent aspects of Sephiroth's character. Kadaj represents Sephiroth's cruel side.