To get this achievement, you have to escape with the survivor Ash with only his three unique perks. You will have to be at least level 10 with Ash to unlock three perk slots.
His three unique perks are Flip-Flop, Buckle Up, and Mettle Of Man. 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. I recommend bringing in a medkit with some add-ons since you won't be able to have Self Care equipped.
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. However, if given the option I recommend escaping through the exit gates. The hatch seems to be hit or miss when it comes to unlocking achievements.