Weapons drop randomly through-out the game so unfortunately I wont be able to direct you to a certain in game spot to get one, I'll try and provide a few pointers though.
You can see if a weapon has an elemental damage bonus by highlighting it in your menu, where the guns stats are listed (damage, accuracy etc.) there will be a small sign indicating if it causes any elemental damage and if there is a small sign, it'll let you know what bonus damage that gun causes, in this case we're looking for a small green biohazard sign. If you find/already have a gun capable of inflicting corrosive damage, just equip it and go kill a bunch of enemies, you can track how many kills you've accumulated by going through the menus, on the quest section press Y and you'll bring up the challenges menu, look through it until you find the corrosive kills challenge and there's a counter there, that's a bit long winded so if you prefer you could always just try and keep count by writing it down or if you're doing it in sitting, keep count in your head.
There are other ways you can get this achievement, if you can't find a weapon capable of inflicting corrosive damage, try and kill enemies which are in close proximity to the green barrels scattered through-out the wastelands, you could alternatively use create a new character and use Lillith (if you're not already), at level 16 you'll be able to upgrade her melee attack and make it corrosive.
Don't forget to check the weapon vending machines too as they do stock weapons with elemental bonuses from time to time.
Finally, lots of people say the achievement is buggy and doesn't unlock when it's supposed to, while this didn't happen to me I know it has happened to a few of my friends, if this happens to you, do not continue getting kills hoping that it'll unlock!, it won't, instead back out and create a new character, it's a pain that you have to do this but it's really all you can do if you suspect that it's been bugged out for you. The bug which caused this to happen has been patched, but I've included it in the solution on the off chance it happens to you! If the achievement did bug on you, boot up your file with the new update and the achievement should pop straight away!
I've linked in a video so you can see the effect the weapon will have on an enemy if you're using a corrosive weapon.
*Credit to Gkthepwner for the video