Must be obtained in a single playthrough. Returning to the title screen, through quitting or death, resets the progress to zero.
Setup:
Pick the first guy as your ship. Hold to fire your primary weapon, release it to move faster, and tap to perform a diagonally forward dodge in the direction you're strafing. This allows you to pass through objects unharmed, with good timing. Occasionally you'll find a power up which you must dodge into to collect. Sometimes dodging will leave a green arrow on the screen - press again to perform a chain dodge, often destroying things in the process. If you ever become a frog (!), your dodge becomes much shorter - prepare for this.
Method:
All the bosses in this game require the exact same strategy - pump them full of lead (or laser) and dodge their attacks with They're by no means difficult.
Some bosses have more than one form / phase - you must kill the entire boos across all it's phases for it to count.
This means to kill four bosses you'll have to reach at least stage 5, which should also be enough for your 200,000 points achievement too!