Is this a crime that the police would handle?

It is illegal to hunt or, kill cats anywhere in the United States. There are exceptions if the animal is posing a danger to people or, livestock. It is better to trap them if possible. Turn them over to authorities to be humanely destroyed.

I've hunted Ferrel Cats for several reasons. Once I had Ferrel cats coming around, eating all My Pet cat's food and picking fights with them. One shot with a 22LR put an end to that BS. Another time I had strange cats coming onto my property and killing my Free Range Baby Chicks. That one I had to run after the first shot and catch up to it to finish the job. Of course if I lived within City limits that would be against the law.

Edit: Of course there's always the possibility of lacing some raw chicken with anti freeze and put out a little treat for the feral bastards, just keep your own pets inside during the meal and tell your neighbors likewise (if they think the cats are also a problem).

What kind of cat. A Lion, a Leopard, a Cheetah.

Here in America you can't shoot a gun in the city limits (some exceptions). Some places even prohibit shooting bb guns in the city.

Killing a house pet would usually, if not always, be illegal.

There's currently a case where a woman died, in her will she wants her dog to be killed and buried with her. Legal experts interviewed on tv have had differing opinions about the legality of it.

Legally, a pet is property and has no rights. But, killing a pet that's not sick, might be cruelty to the dog.

You are not allowed to kill people's pets.