Couple of tips for this achievement.
- Aim for the earlier levels (Either 1, 2, or 3). These first 3 levels (Outer Space Panic, Neptune's Crypt, Goosebumb Gorge) have the lowest enemy requirements to unlock the boss. The 4th through 7th levels raise the requirement and Level 8 has the highest (50 enemies). Those first 3 levels should be more reasonable than the others.
- While Level 1 has the lowest enemy requirement at 15, that might not make it the easiest. The snake/chain enemies (ones that have multiple pieces that all fire shots towards you) are multiple enemies in other levels. In Level 1 however, it is a single enemy and thus only counts as one towards unlocking the boss. Up to you however and how comfortable you are.
- No Shields or Bombs. You can't use any for this achievement.
- For the power-up drops, I recommend aiming for the extra lives. If you're not firing extra shots all the time, you can wait for it to cycle to the extra lives. Keep in mind the life is after the reverse directions "power-up". All the others won't really help you for this one.
- Out of the 3 bosses, I actually found Neptune fairly easy, but the Cat on Level 1 also isn't terrible. I wasn't a fan of the graveyard boss as his vertical movements got a little unpredicatable at times and thus caused some unfortunate hits at times.
- Move in small circles. I found that the circular motion helps dodging shots (especially from the enemies firing a fanned out shot patterns). You can weave better knowing this. By the time I played through the game on the second time, my dodges were a lot better as I got more accustomed to the enemy patterns and recognizing that although the enemies look very different, there is a very limited set of types.
With some practice you can do this. Given the ability to stock up bombs/shields for the rest of the game and any of the other achievements, I'd estimate this is the most difficult achievement in the game.