This achievement involves fully killing (not hurting or putting into an executable state) 10 aliens using the various explodable items in the environment such as barrels and cars. Just to clear up, this doesn't include destroying the vehicles they are on or using your own explosive weapons - though that should be obvious.
The first decent place to start using environmental explosives is late into the Duke Cave. Just before you reach the turret you'll be in a circular corridor with around 5 enemies and three or four explosive canisters. It's easier to kill enemies by picking the barrels up and hurling them at the aliens just be wary of getting shot whilst holding them - it drains your ego nearly completely on any difficulty.
Your next best opportunity would be the pit where you fight a large number of Pig Cops and Alien Commandos after you've traversed the roads and witnessed the death of "Poor Leroy" lol. There are a few barrels around though it does get hectic and they can be destroyed by the RPG-wielding Pig Cops that spawn later into the battle.
The best place (probably the last time you'll need to actively try for this achievement) is on the crane lift in the building site on the Duke Dome level not long after killing the Battlelord in Las Vegas. Keep watching for enemies in the car park that you are lifted past, there's plenty of opportunity to kill 2-3 enemies with the kindly placed canisters per battle. There's probably enough just to get the achievement from this one area.
It's worth noting that this tally, along with most of the achievements in Duke Nukem Forever, keep your score regardless of whether you die or start a new game - very useful if you need/want to restart the checkpoint at the car park crane lift for the last few you need. Hope this helps :)
Edit: It's best to use the Ripper, pistol or laser at the crane scenario. Don't use explosives else the barrels won't be responsible for the kills, your gun will! Hopefully goes without saying but if it helps one person ;)