To get this achievement, you have to escape with the survivor David with only his three unique perks. You will have to be at least level 10 with David to unlock three perk slots.
His three unique perks are We're Gonna Live Forever, Dead Hard, and No Mither. You will start with tier 1 versions of these perks but I recommend trying to get them to tier 3 to make things easier for you. After level 10, the Entity will start to eat away at the bloodweb, so make sure you go for these perks first when they show up.
Equipping items, add-ons, and offerings does not void the achievement. Unfortunately No Mither makes medkits useless for yourself so don't bother bringing one in.
Thank you SKORPIONIC for the heads up: “Also, seems like you have to escape through the gates, not the hatch. My first time with the 3 perks I used the hatch and nothing.” Thank you He a Good Boy for confirming. Escaping through the hatch no longer voids the achievement.