If my opponent win a military challenge against me can I kill the bastards elite for claim before it's passive effect takes place or do I have to give up control of the card first?

Claim happens well before passives. You can kill him for claim before he changes control.

He will still, though, change control in the passive step while moribund. This might matter for something like Parting Blow.

You are bringing up Parting Blow because at that point it is no longer your character that is leaving play is it? What's funny is that means your opponent could play their own Parting Blow after winning a MIL challenge against you.

You are bringing up Parting Blow because at that point it is no longer your character that is leaving play is it? What's funny is that means your opponent could play their own Parting Blow after winning a MIL challenge against you.

Yes, exactly.