HELP! Why did Klein shoot the wife in the car?

As far as I can make out, I think he really did believe she was already dead ( from causing her deliberate crash ) and he shot her to solidify validity to his story that Silien shot her.

Because he caused the car accident that killed her; by pretending that the criminal shot her while she was driving it's as so the car accident happened because of him shooting and killing her therefore making him not responsible of anything.