It would be tough to hunt a cat. Normally you hunt them with trained dogs. Do you live way out in the middle of nowhere? I would think poison would work..... but, you can't do that of course because it is probably illegal to kill them where you live. Plus, you would run the chance of poisoning somebody's dog.
A neighbor of mine got prosecuted for shooting a skunk in his corn crib. I use a have-a heart trap, those kind that don't injure the animal. I then load it into my truck bed, drive a few miles from home, shoot it dead and toss it on the road like a car hit it. Coons too! As long as skunks can't raise their tail they won't spray.