This achievement isn't hard or annoying but it's specific in what the requirements are so here's what I found by screwing up 2 runs before I got it.
Don't use flashbangs/melee/nades they do actually void the achievement, the game literally means only use your 1911, if someone gets too close remember Zombieland's Rule #2: THE DOUBLE TAP do NOT melee. Keep this in mind at all times, especially when climbing up the ladders near the end when everything is on fire, enemies spawn in randomly, so stay frosty.
Don't pick up any other weapons either, disarming the tripwires is fine however. I'm actually not sure if this is a requirement but after screwing this achievement up twice I wasn't doing this mission for a 4th time, the 2nd run I picked up a FAL on accident and didn't get the achievement.
Take it slow, aim for the mid to upper chest and headshots, you will need the ammo even on recruit, if you happen to run down to about 7/21 (7 active, 21 reserve) and if you die/restart checkpoint you get an ammo refill (at least on Recruit difficulty).
When you get to the MG's nest, run then slide to the left (making sure to kill the guy there first) when it overheats and prone. then mantle over the wooden wall you've just taken cover behind, prone again then run on the 3rd overheat, that thing shreds even on recruit difficulty, be mindful the gunner has a friend sat beside him with a shotgun.
Maka has a video showing how to get this achievement: