This will be a tricky achievement since its partially dependent on luck (the waves are quite random). To make this as easy as possible, start a private Firefight game. You can change the difficulty to easy and still get all the achievements.
There are a lot of other setting you can mess around with but I don't know if they will disable the achievements or not (they will disable leaderboard scores). Changing these settings will disable achievements, so don't bother with them.
In Firefight your objective is to keep the Terminus (the big building in the middle) alive as long as possible. For this achievement, the Terminus cant be damaged for 15 waves straight. The Terminus is surrounded by 3 spires (no clue what they do). These spires can be attacked without breaking your streak. The streak will only reset if a message shows up about the Terminus taking damage.
This solution is if you want to try this achievement solo (coop is probably a little easier).
The leader I chose was Jerome. In your base you want to build 2 supply pads, 2 generators and a field armory. Get some marines out and build Jerome as soon as possible. Your best bet is to make Hornets since they are fast and do quite a lot of damage. Get the 2 minibases up fast so you can build some extra sets of turrets. The Terminus itself is surrounded by 6 turret build sites. These should be build after you have a stable economy.
When you have a decent amount of Hornets, try to expand to the next 2 sets of minibases. This will increase your unit population and allows you to build even more Hornets to help you defend the Terminus.
Make sure to pay close attention to where the enemies will come from (the blue line on your minimap) and what type of enemies is showing up (you can find this in the top right of the screen). If the enemy group is a bandit raid or direct air, get your army back to the Terminus because the direct air will go straight towards the Terminus. The bandit raid will try to steal resources from your bases, however, when they fly away from your base, they might get close to the Terminus and attack it.
On easy nothing is really that hard to kill. The only things that can screw you up are the air waves. Also, it appears that sometimes a stray rocket can fly past your units and defenses and damage the Terminus. There isn't much you can do about this.
I would highly recommend to get this achievement in the first 15 to 20 waves because afterwards, the waves will be to big and strong to quickly destroy once a direct air wave appears.
As some people have mentioned, this has to be done in the first 15 waves. It will not unlock past wave 15. This might be a bug, but who knows.