Place the kitten in the cat house to turn it into an adult cat. The process takes 24 hours. Don't forget to leave enough food and water in the cat house to stop your pets from becoming exhausted. You can also feed your pet Cat Treats and milk to restore its stamina.
You can upgrade the cat house to level 4 and have up to 20 cats in your castle.
The design can be found in the Traitors' Catacombs and well as obtain them by defeating Aissa. The parts for the cat house can be found at 5 Skull locations and after defeating Rogvold.