Nine Lives

Whilst ignoring Auriaya, single target the Feral Defender and kill it. It will respawn an additional 8 times and upon killing it 9 times, and then Auriaya, you will obtain the achievement. You can check progress by counting the black pools when the Feral Defender dies or by checking it's Feral Essence, after 9 kills it will no longer respawn.

The Feral Defender spawns during the fight with Auriaya--you'll see her start the spell "Activate Feral Defender", and then it'll periodically resurrect itself after that. It has no relation to the Sanctum Sentries the boss starts the fight with--those you can murder with impunity, though if you are doing this at a lower level than cap, be aware killing them makes the boss stronger.

Unfortunately, none of these beautiful kitties are tameable.

Ok, If you want to get both achievements Nine Lives and Crazy Cat Lady, this is completely possible:

Information

1. 
   
2. For Crazy Cat Lady you need to kill her without killing any of the cats walking around with her ([npc=34014])
3. 
   
4. For Nine Lives you need to kill the [npc=34035] 9 times, this NPC is NOT related to the cats she have walking around with her, this NPC will be a summon from the boss
5. 
   

Steeps

1. 
   
2. Start the encounter with 0 DPS to anything, cats or boss
3. 
   
4. Just wait for the [npc=34035] summon and after is done, start killing him 9 times (You can count using the pools in the ground or with the buff he have showing how many lives are pending, but after the 9 times he will be gone so no big deal if you don't want to count)
5.