There was a homebrew "catkin" race that I slightly altered with the DM's permission. Basically, the story was that Sprinkles used to be an old Wizard's pet that, during some arcane experiment, got hit by a supercharged version of the "Awaken" spell, which endowed the feline with superhuman intelligence (score of 18 at lvl. 1). He subsequently escaped the wizard's study and set off on his own, intending to surpass the old master by using his new-found spell-casting abilities. He behaved more or less exactly as you'd expect a talking cat to: self-important, arrogant, and generally pompous and big-headed. He also insisted on being referred to by his "wizard name:" Thermidor the Magnificent. He was great fun.
No, there's no rules stating that you can play as a cat, but an appropriate backstory can make just about anything happen.