Gears of War 4

If you did the five maps before the update then you should have automatically unlocked this achievement upon starting up the game.
If you did less than five maps you should see 20% gained progress in the achievement app for every map you have completed (i.e. if you did three maps then you should see 60% completed progress).

Playing in normal queue on any difficulty and playing in private count.
I did a match in the Developer Playlist and I did not gain any progress. Maybe it just glitched that run, I don't know. (UPDATE: It appears I gained progress from the playlist once that map became available for everyone.)

Each DLC map will only count once. If you decide to do multiple speed runs on Security, only the first one will give you progress towards this achievement.
You can play with any amount of people as long as it's at least you and one other person. (Credit to HoHe69 for this correction.) However more players would be ideal but the game will scale the amount of enemies based on amount of players.
If someone gets disconnected then that disconnected player won't get credit.

Season Pass is not required to complete this achievement however you will need to know someone that has the Season Pass and they will need to host the math if you plan to do this in private. Really no reason to do this in matchmaking as you get no say on the map and random players will just screw up the game or leave after wave 20. I suggest setting up a session on this site or using the Looking for Group feature on the console itself.
DO NOT BUY INDIVIDUAL MAPS! While you can just buy the maps you want, you'll end up spending half the cost of the Season Pass for only a fraction of the content.

Maps that will not count towards this achievement:
-Dam
-Fallout
-Forge
-Foundation
-Gridlock
-Harbor
-Impact
-Lift
-Reclaimed
-Relic

Maps that will count:
-Avalanche
-Blood Drive
-Checkout
-Clocktower
-Dawn
-Diner
-Dry Dock
-Glory
-Hotel
-Impact Night
-Old Town
-Raven Down
-Reclaimed Windflare
-Rustlung
-Security
-Speyer
-The Slab
-War Machine
-Any other new maps not in the first list

Now any DLC map should be easily doable for anyone of any skill level set on Casual but just in case I will include another list for extra easy or exploitable maps.