Welcome to Exo Survival, the co-op mode that you most likely remember from mw3 (which sledgehammer made). As you may already know, this achievement ties into the hidden zombie mode which unfortunately isn't unlocked straight from the start.
How do you unlock the Exo Survivor achievement then you might be asking? Well it's simple. You must play enough rounds of the co-op mode to unlock it!
Let me explain; The map we are looking for is known as Riot (the prison map) and is ranked as a "TIER 4" Map which means we have to go through the previous TIERS to unlock it. From what I've seen the rounds count in private match also so if you have some buddies on, invite them in and start beating as many rounds as possible on each map and you will unlock it.
TIER 1:
Map: BioLab
Map: Retreat
Map: Detroit
Map: Ascend
You must complete 50 "TIER 1" Map rounds to unlock "TIER 2".
TIER 2:
Map: Horizon
Map: Comeback
Map: Terrace
Map: Instinct
You must complete 75 "TIER 2" Map Rounds to unlock "TIER 3".
TIER 3:
Map: Greenband
Map: Solar
Map: Recovery
Map: Defender
You must complete 100 "TIER 3" map rounds to unlock "TIER 4"
TIER 4:
Map: RIOT
Now we have "TIER 4" unlocked, jump into the map Riot and start playing. You must overcome 10 rounds of normal enemies and once round 11 has come around, you will have started the Bonus Round. All you have to do now is survive and then once you get a countdown timer on your screen, run towards the extraction zone and after a cutscene plays, the achievement will unlock.
If there's anything I should improve please let me know! and if downvoted post why as it helps me write better guides!
Edit 1: I have posted a method over here which will help with all Exo achievements and a way to reach riot faster.
Edit 2: If you have a friend or find Somone with the achievement and they have the map riot, you can skip all the above and get the achievement that way but you won't make any progress towards the other achievements (50 matches, Classes, Flip a Map)