This might be one of the most challenging achievements for sure, but don't fret, I'll give you tips on how to get it:
1) Focus on upgrading your MAX HP, this might sound counter-intuitive, however if you are the kind of guy who don't wanna bother avoiding the projectiles at the last boss stage (I'm referring to stage 10, Stage 9 is a bit on par IMO, but it depends on how good you are at tanking a boss).
2) If you have found the GapMinder Repro in a shop, I highly recommend you buy it, as it is basically like a safe ticket in case you already managed to defeat the final boss but you fell into a pit with one HP left.
3) When you reach stage 10 and defeat the final boss, jumping from one platform to the next is easy and you can take a full minute or two before jumping, there is no rush, you will encounter Aces and Ninas along the way, the goal here is to let them damage you to the point where you have one HP, if you have already one HP keep going.
4) BEWHERE THE LAVA PITS!, as soon as you progress enough in the map the camera will focus a bit on an inside part of a building, ahead you will find about 3-4 pools of lava all close to each other
Note: You may use the lava pools to reach 1 HP before reaching the space rocket.