Man, turns out this achievement is a little more difficult than it appears. So like the guy who posted the first solution here are the basics:
1. Beat adventure mode. This will unlock the other game modes such as Iron Frog and of course the one that concerns this achievement Boss Rush.
2. Beat Boss Rush
3. Do not die. This is obviously the key as these are the requirements set forth by the cheeves description.
Now obviously the hard part is not dying (losing) and if this should occur to you, you will need to exit boss rush and start over in order to ensure that you get the achievement.
For me the hardest part came when i finally got to the stone head looking guy. His attacks are fast and he has two minions that also shoot attacks at you that will slow your shot down. The method that I found works best is to limit the number of shots that you fire at him. The stone head only needs to be hit a total of 5 times before he changes into the bat guy and his attacks while stronger are much easier to avoid. The way you limit the number of shots fired at the stone head (and this applies for the remaining bosses) is to only fire when the stone head is just about to approach a section that you are able to clear. So for example if the stone head is all the way to the left of the screen and two yellow balls are together right in the middle of the line and you have a yellow ball up in the cannon, fire it and clear the two yellow balls. Just before the head passes over your cleared space fire a ball up there and BAM! youve got him once. repeat this 4 more times and you are onto the bat guy. The only other time I would fire a shot is to clear some balls to cause a chain reaction or to get the balls away from the mouth that causes you to lose.
Now once u have moved onto the bat guy continue the same strategy as above by only firing when absolutely necessary and you will move onto the final boss the frog. The first 4 or 5 hits are very easy because he will follow you as you move left and right. When he starts to move the opposite direction as you the strategy from the last two bosses becomes key again. Move your frog to the middle and only fire shots when you can clear the two balls directly above you and then fire a second shot at the boss.
It took me two tries using this method and my friend did it the same way in two tries. Sorry for the really long explanation but it will work. :D I wish i had a youtube video to help explain better but i think the length of this explanation should do. If anyone wants to send me a link to a youtube video that does what ive described id love to put it up. Cheers and good hunting!