This starts in Chapter 2 at the end when inside your house. There will be cat sounds so shift to Living world and interact with the back door. You can choose any colour and name for the cat. Now go to the kitchen and on the cabinet the option to Feed the cat, next on the couch use the pillow as a cat bed. You can now take the first photo of the cat.
Chapter 3 at the end when your place is trashed the cat will be on top of the fridge. Take a photo.
Chapter 4 at the start find the cat in front of your bed. You can take a photo and finally pet them for the achievement.