I was able to do this by focusing on traps. Traps can get 175% increased damage on frozen enemies, 100% on burning enemies, and 100% on poisoned. If you have a way to freeze, burn, poison (or all 3) enemies and you have those mods on level 20ish Crossbow traps, you will melt the boss without using anything else.
If you have unlocked the permanent upgrades that give random weapons, you can't get the achievement with those. Players with theses upgrades can still find the original gear hidden in the walls under and next to the bridge you start the game on.
TLDR: make sure you're taking the actual starter gear that would spawn on a brand new game.