The majority of all the enemies you'll run into no matter what. But the few that people miss are:
The flying blue bird -- Location: Close to the end of Chill Man stage, you'll see them at the top of the screen dropping bombs.
Single-decker truck, first half of Nitro Man stage. Use chill spike to kill it.
Double-decker truck, second half of Nitro Man stage. Again, use chill spike to kill it.
Search Light, Dr. Wily 2. There's only three in the stage, two in the beginning and one about halfway through. Just shoot it with megabuster.
Flames (from flame generators) -- Location: Solar Man stage. Use water shield to destroy them.
Also easily missed are the enemies that spring out of the ground and spray bullets. Destroy them with Commando Bomb, (Thanks to sacredskullkid for the tip!) or Chill Spike (Thanks to Swordfish00830 as well for the tip!).
Thanks to KMetalmind for this tip: It should be noted too that if you leave any enemy, and you finish the game, you must kill that enemy and finish the game again. This happens because you can´t save once you´ve beaten the game, so enemies from Wily´s Castle (or at least the drills located in Wily Castle 5) aren´t counted for the achievement. If you managed to save between levels, those ones will count. The achievement pops up while killing a drill usually, as it´s the last enemy in the game (final boss isn´t needed as far as I know).