To get this achievement you must kill 25 enemies (skags, bandits ect) with weapons that have a corrosive element.
These weapons have a green hazardous chemical/materials (HAZMAT) sign in the inventory menu.
Green numbers will come off enemies if they are impacted by corrosion. Corrosive damage is like other elemental damage in the game in this is causes damage over time. The enemy does not have to die as a result of the elemental damage however.
You can check your corrosive kill count in the challenge menu, go to the quest screen and press Y.
You can also use barrels for this achievement. Corrosive barrels are green.