This is a very hard achievement to obtain, one definitely for your trophy case.
The key here is PATTERNS. There is a certain style and pattern for the bosses on each level.
Each level at the start gives you cannon fodder to shoot up, reach a certain number of kills, and you'll face the boss for that level. It sounds VERY daunting to do this 100 times, with increasing difficulty each time.
Lets get the trivial bits out of the way first. Set your movement rate in the E4 mode settings to the maximum (fastes) setting, and set your shooter to shoot in the direction you are aiming, NOT the 4-way shooter.
The "cannon-fodder" part of every level is always approached the SAME way. All you need to do is hold the shoot button and constantly twirl your right or left stick in a continuous circle, your choice of direction. You're gonna need to do this at least 100 times so find your rhthym and make this thumb action your second nature.
Now, like previously stated, each level, starting from 1-5 has its own unique 'boss'. You'll know what boss to expect when playing, but be wary of the BIG boss. This guy is best taken out by holding your shield from the very start and moving in next to the core while holding the trigger, thus filling him with hot lead and killing him before your shield runs out (hopefully). NOTE: In the starting stages, you can develop and technique to 'evade and shoot' which is necessary if your shooter hasnt reached its MAX level yet. But dont rely on this towards the 60s because there will be a point where its IMPOSSIBLE to 'evade and shoot'.
Towards the final stages, you'll need more than 1 life to use on one BIG boss.
The holy grail of climbing the level system of this mode is to NOT WASTE ANY LIVES on ANY of the "sub-bosses" throughout the levels. Even in the 80s and 90s levels, i managed to survive the sub-bosses whenever they cropped up.
Nothing I can write can help you much with them, its your learning curve to learn their pattern, and develop a technique to evade and despatch them in your style and your comfort zone.
Remember, avoid deaths, every life will count so dont waste them. Your shooter's power will HALVE if you die, so you will suffer heavily if you dont maintain the MAX level 20 power shooter in your arsenal.
I would recommend a video, but all the ones ive seen arent really helpful.
Above all, take your time, be patient, learn from your mistakes (write down where you went wrong every time) and dont feel pressured to keep trying and do this on the same day. Give yourself a break, and it will come naturally.
Boy o' Boy did it feel good to unlock this achievement, especially because it was my last one for the game. Good luck!