Here, Kitty Kitty...

This is easiest way to solo if you can stand the damage for 5min.

First bind Interact With Target to a key you like.

There are four rats so get one from upper floors. When you enter boss room turn right. Now you should be in corner next to brazier and temple rat is other side of the wall.

Pull boss and aggro 4 cats and use temple rat on the ground. You can kill the cat which ate it. Now you can get more rats every 1 minute by doing /tar temple rat and spamming interact with target key. Don't forget that you might target upper floor rats if rat on your floor has not spawned yet.

I was confused and didn't understand this the first time I read it, so here is some clarification. You need to be targeting the rat that stands in the corner right OUTSIDE the boss's room, then stand on the other side of the wall (INSIDE the boss's room, next to a group of cats) and spam your "interact with target" key while you have the rat targeted and you wait for it to wander close enough. This allows you to grab the rat through the wall and bypass the annoyance of trying to do it just outside the boss room without having the boss reset.

This is a clarification of the process presented by tyuiop. That process worked well, but I found it a little confusing. This specifically applies to level 110 characters soloing this achievement (or any level that can survive indefinitely for this fight).

1. 
   
2. Bind a key to "Interact with Target" (Found in the "Targeting" section most of the way toward the bottom. I used the comma key.)
3. 
   
4. Track the achievement Here, Kitty Kitty...
5. 
   
6. On the way down to the boss, grab a [item=69724] from anywhere upstairs from the boss, not the one right next to the boss room. (They don't stack, I'm not sure why they show that they stack to 20 on the item page, don't bother collecting more than one.)
7. 
   
8. Pull the boss without doing damage. (Walk into her for example.)
9. 
   
10. Damage the boss VERY CAREFULLY using whatever wimpy attack you have. (Remove armor if you have to so you don't kill her.)
11. 
    
12. Your goal is for her to wake up all her cats. (I found I couldn't pull the cats without them dying, so I needed the boss to do it for me. You can skip this if you can pull the cats yourself.)
13. 
    
14. Now, with all the cats active, use your [item=69724] from upstairs. (You should now have 1/4 fed.)
15. 
    
16. Walk over to the green barrier and click the [item=69724] (or wait for it to spawn in 30-60 seconds, and then click it).
17. 
    
18. Walk around the wall to the left of the green barrier toward the corner where the brazier is.
19. 
    
20. Spam your bound "Interact with Target" key until it says "You receive item: [item=69724]"
21. 
    
22. Feed the cats again by using your [item=69724]. (Repeat steps 8 through 11 until your achievement tracker shows 4/4 fed.)
23. 
    

If all goes well you never needed to /target anything and you should now be able to kill the boss and get the achievement.