If the cat has a tendency to drink the water I'd avoid the bleach (a small bit wouldn't kill it but it might lead to some horrible hacking up of hairballs and hidden piles of cat vomit) -- use vinegar instead, it should have the same effect on the bugs and if the cat doesn't avoid it for the smell, it won't be harmed by ingesting it either. It needs to be cleaned with detergent every time, or else it will become an algae fest over time.
EDIT: In the last sentence I'm referring to the bowl, not the cat
EDIT: In the last sentence I'm referring to the bowl, not the cat