Why is Xenon always trying to kill Supergirl in the comics ?
I mean, I know he's a villain, that's what he's supposed to do, but why does he hate Supergirl specifically ?
We learn in Supergirl Vol 4. #78 that at some point in the past, while engaged in general villianry Xenon was confronted by Supergirl. At the end of this encounter, she shoved him into the void between universes where he was stuck for uncounted eons. He took that quite personally and, on securing his release, has made it his personal business to travel through time and space throughout the multiverse killing all Supergirls everywhere.
In addition to fulfilling his personal vendetta, his long-term goal (given that he doesn't know which of the countless Supergirls were responsible) is to murder them all, thus preventing his imprisonment from ever having taken place in the first place.