If you are struggling to get to / beat this boss, do yourself a favour and from level 3 onwards (the restart checkpoint) concentrate on shooting the power up ships that float across the top of the screen on most of the levels, rather than just trying to quickly clear the levels of enemies.
Ideally, you want to try to get all 3 of the power ups (speed up, power up & shield) by the time you get to level 6, but your best friend is really the 'S' (shield) power up as that is essentially an extra life. Once you have the shield (and ideally both of your starting lives) you should be able to get this achievement without too much frustration.