MEGA MAN 10
12 Achievements
200
12-15h
DESTROYER
Defeat 1000 enemies.
20
How to unlock the DESTROYER achievement in MEGA MAN 10 - Definitive Guide
Anyway, that being said, the game counts your 1,000 kills from game to game, unlike Mega Man 9. I got this achievement near the end of my third playthrough. This means that at no time do you ever really need to think about getting this achievement. It should come naturally.
If, for some reason, you insist on focusing on just this achievement, you can shoot the drills in the last stage, as Shi Kiseichu has mentioned. This scenario seems likely if you're ignoring some of the harder achievements (like Mr. Perfect or Superhero) and have only played through the game once or twice.
Just defeat 1,000 of any enemy type and the trophy will unlock as soon as that last enemy falls. There are a couple of good farming areas, but the one I used for a bit was in Nitro Man's stage: before you climb the first ladder, equip Blade Man's weapon and dart back and forth before two enemies that dash at you rapidly. Blade Man's weapon will kill them in one hit (if you kill the head and body at once) and they are pretty generous in Screw and Weapon Refill Bubble drops. The trophy is cumulative, so you can accomplish this over multiple playthroughs (confirmed by Smirnoff103). You can earn this trophy by playing as either Mega Man, Protoman or Bass.
This achievement stacks up as you play through the game. You will need to run through the game several times to defeat 1000 enemies in total. If you have other achievements to clean up this would be a good time to do so.
You will most likely unlock this on your third playthrough. Just shoot and kill as many enemies as possible.