There are 4 different survival modes you have to play:
Survival: Shadow Warriors
Survival: Masked Warriors
With Darkness, Shadow Warriors, and Masked Warriors Survival Modes you only have to reach Round 7 which isn’t too hard. Once Round 7 starts the achievement unlocks and you can just quit out if you’d like.
With Survival: Horde you have to reach Round 21. By doing so you will get the “Zookeeper” and “Extreme Survivalist” achievements.
Tips for reaching Round 21:
Make good use of your dodges. Don’t get crowded by enemies as they will drain your health. In Survival Modes there are NOT any health relics so you will not be able to replenish any health that is lost.
Lure enemies away from the group. Faster enemies will follow a lot quicker giving you time to take them out while the slower ones catch up.
Make good use of the relics. There are two in this level: Infinite Powers (blue orb) and Mega Damage (yellow/orange orb). Mega Damage is great for bigger enemies or taking out a big group. I used a lot of powers so Infinite Powers helps a ton.
Use the air ducts especially when trying to escape and when fighting the giants. I would lift up on an air duct and perform a power attack and alternate between air ducts. By doing this I defeated the giants without being harmed.
Time Stop is your friend. I used this many times, especially with bigger enemies. Just use Time Stop right before you attack to ensure you don't take any damage. This works really well with the Infinite Powers relic.