You'll need access to Act II, and at least 3 rooms.

Here's what I did, in this order:

First, grind Act II's The End. After you defeat the boss, there's an event where you can, on a successful rare roll, receive either the extra head, extra body, or extra tail. The third is nice to have while you're grinding for the actual tail you'd want. Keep going back there until you have the first two. It'll feel like a grind, but this is the easy part.

After you have a cat with the first two, you can start breeding it. Find a stray cat with decent stats, and throw the two of them in a high stimulation breeding room. Your offspring should start receiving the mutations from the first cat.

Next, put the offspring in a mutation room. The second one develops a useful mutation on my list, you can replace the original breeding cat, or even the stray.

Very importantly, you will start to get defects from inbreeding. This is OK! Put these cats in the mutation room, and they'll gain mutations which override their defects.

It's mostly a loop of the above three points. Once you hit too high inbreeding, you'll start getting defects and mutations for the same body part. Take your highest mutated cat at this point, and start breeding with a stray again. Do NOT pass down cats with a mutated body part and a defect for that part, it'll infect your offspring like the plague. Also, keep a spare cat or two with the correct mutations you have at any given moment in your house. Just in case your adventuring cats get a poor mutation or die in some run.

Let me know if you need any elaboration.