Where do I go to change a pet profile theme/layout?

Hover over the Personal Tab, click on Pets, then click 'Edit Description'.

Thank you, but I was asking about using backgrounds. Can I do that?

Of course you may use a background on your pet's page - so long as it doesn't contain anything which violates the T&C.

You can look over the Profile Discussion forums for do-it-yourself coding, you can go to the Profile market forums to commission one, or you can look at sites like SubetaLodge and SubetaHQ for premade ones.