Kadaj is one of three beings created by
Sephiroth as manifestions of "his will". He and his "brothers", driven by Sephiroth's will and their own personal obession, sought out their "Mother": Jenova's head. Kadaj represents the bulk of Sephiroth's personality.
Having Yazoo abuduct the children with Geostigma, a product of Jenova's influence in the lifestream, he intended them to serve as bloodhounds
to find Jenova's head. He even used the Bahamut materia to hold the other at bay until he learned Rufus possesed Jenova all
along.
Through asborbing Jenova, Kadaj became Sephiroth, whose intent was to use the Geostigma to choke the lifestream and use the planet to start over again on another planet,
much like Jenova did in the past. This did not happen because Cloud defeated Sephiroth, mortally wounding Kadaj in the process. Hearing Aerith's voice and assuming her to be his "mother", Kadaj was freed of Sephiroth's influence and faded into the Lifestream.