Cats are found in villages. They can be tamed and befriended by feeding them raw cod or salmon. When you find a cat get, within 10 or so blocks and stop moving, this includes jerking the camera, as sudden movements will cause them to run off. Then just feed them with L2 until hearts appear.
I CAN confirm that turning off autosave, saving before a tame, taming, quitting, and reloading does in fact work. I found a village with 4 cats and tamed all 4, quit, and retamed them 5 times and the trophy did indeed pop.