Ikaruga

Since you can earn these achievements on any difficulty, I'd suggest just playing through on easy.

For this boss, you have to open up at least 1 of 3 series of gates to attack the center of the boss. I recommend focusing on just 1 series of gates. Open each of them using the opposite color to make it go faster. Once you have all the gates open (I think there are about 4?) in one series, try to position your ship inside the boss.

You can hang out inside the channel left by opening the gates and pummel the boss. This saves you the trouble of dodging the streams between the boss and the outer ring. If you're good enough with your movements, you should be able to just circle around with the channel and avoid dying.

If he starts to shoot the powerful beams from the center things may get a little harder. It'll push you outwards so try to fight it but be cautious of the sides of your little channel.

On easy, you shouldn't have much difficulty and within a few tries, you should be able to do this without dying.

Defeat the boss of level 4. Keep shooting the center. If you have enough credits just sacrifice lives and stay in the center. (one credit gets added per hour of game play; 7 hours of playtime unlocks infinite credits. Play time can be checked from the menu.)

To unlock level 4 you must make it there starting from the beginning. Simply selecting level 3 and beating it will not unlock the level.