I got this achievement in Stage 5 on a regular playthrough (not the arcade mode, though it might appear there as well). The enemy in question is the large ship near the bottom of the screen that has two white finger guns which take a decent amount of firepower to take down. It slowly drifts away from you during the battle, so if you aren't quick enough in killing it, you'll miss this. I assume the worm is guy steering the ship, as he looks... wormy. I didn't have to hit him, just the white hands that were shooting at me. Once I killed both of them, the ship seemed to lose functionality and head down off the screen, indicating it was destroyed, and the achievement popped.
Just to point out, you will encounter a similar ship on an earlier stage (Stage 4, I think?) that has a few of those white hands that shoot at you, but when I killed that one, the achievement did NOT pop, so I think the one on Stage 5 is directly tied to this achievement.
Using a pilot/ship with higher damage will make this easier, and using the flame super ability (not sure what it's called, but it's the yellowy one that looks, well, like fire) will hit both hands pretty easily.
Dodging the hand's projectiles while firing can be a bit tricky, but this is far from a difficult achievement if you focus all your attention on this enemy.