In order to unlock this achievement you will need to kill 25 enemies with a corrosive weapon. If you are wondering if the weapon you have does corrosive damage just go to your inventory screen and in the area that shows the damage, accuracy, etc. of the weapon you will see a green haz mat sign just like the one in the achievement pic as well as a multiplier of x1, x2, or x4 which indicates the chance that the effect will proc. You can also do corrosive damage by shooting green barrels that are close to enemies. I've noticed 2 different green barrels, 1 that explodes and one that splashes acid. Make sure it is the 1 that you see acid splash out of. If the enemy dies as a result of this acid barrel it will count toward this achievement as well. These barrels show up within the first 10 minutes of the game. Basically if you see green numbers coming off of the enemy you are doing corrosive damage.
I can confirm that this achievement can glitch. I had over 500 corrosive kills with no achievement. You don't need to clear your hard drive of all game information you just have to start a new character and re-do the achievement with the new character.
You can also track how many corrosive kills you have by going to the quest screen and pressing "Y" which will take you to your challenge screen. Then simply scroll down to where the corrosive kill counter is.
EDIT: MulletGuy reports that a recent patch has remedied the glitch associated with this achievement. If you met the requirements previous to the patch with no achievement, you should receive the achievement upon updating.