Solasta: Crown of the Magister

EDIT: Place of Ice update has made this easier.
If you pick one damage achievement to go for early, make it this one. Lots of late-game enemies are immune to poison. Specifically, Undead, elementals, and constructs will be immune to poison damage.

Applying poison to a weapon uses your action, and only lasts a minute. So it's hard to use in combat or before combat. Arrows can be made pre-poisoned, so use those. For the achievement tracking, you need to deal the last hit point with poison, so the target needs to survive your weapon's base damage and then die to the poison.

Instead, you'll focus on getting this with your wizard, sorcerer, or high elf. Pick the poison spray cantrip, and at nineth level, grab cloudkill. Both of these have problems. Poison spray requires you to be within two squares. If you take the powerful cantrip feat at 4th level, it'll always do half damage. Cloudkill creates a mist, blocking line of sight, and causing damage each turn to everyone within, including your allies. Unlike other spells, cloudkill will move away from the caster each turn.

You must use poison damage to finish off 20 enemies in one playthrough.

I found the most consistent method to be poison spray cantrip on low level foes as things like poison arrows can kill with the initial piercing damage and it then does not count.