Advent Children (Complete) fails to explain it properly at all. There's a novel (Final Fantasy VII Lateral Biography Turks -The Kids Are Alright-) which sheds some light on Kadaj's origins. He was apparently (SPOILER)"born in a pond of blackened water near Nibelheim" and becomes a Sephiroth Remnant - equivalent, as far as I can tell, to the other Sephiroth clones - and then there's this bit: "only to end up diving into the Lifestream and dissolving after Evan kicks a still-living remain of Jenova into it." Yeah. Take that as you will.