Is it legal to bury a cat in your backyard?

It depends on where you live.

In general it's probably okay providing you comply with certain guidelines. For example, many locations specify a minimum depth and have regulations about how near the grave can be to wells, flood plains, and/or wetlands, and how many graves can be within a given area (probably not a problem unless you are trying to run a pet cemetery).

The best thing to do would be to look up the regulations for the disposal of animal carcasses in your state or municipality.

It should probably also be noted that burying a live cat (you didn't specifically say the cat in question was dead) is probably illegal nearly everywhere due to animal cruelty laws.