How to unlock more backgrounds

The only thing you can do is get better food and items. The cat stay more time in your yard and that means more chances or getting gold fish. Usually, when a cat can give you more than 50 normal fish, the next time he can give you gold fish. Also, there are rare cats that usually give you more gold fish than others.

Just try to have the best food and items.

To get new cats to appear you need to put out different items as they attract different cats, your food bowl also needs to be filled, if it's empty no cat will appear, to find a list of what cat like which items check out this Link.

This provides a list of all cats and what items are needed, in the related section on the bottom of that page you should find the link to part 2 aswell.