Blue Tighten is the final boss and you go right into this fight when you select level 10, "Blue Tighten Battle". For the achievement to unlock you just need to defeat the boss without dying. Here is the method I used:
It is actually pretty easy. You will be in a robot suit and just need to move your cursor around and shoot any debris the boss throws at you. He will eventually grab a column and try to hit you with it. Just destroy the column out of his hands before he gets the chance. Repeat with 2 more columns and a portion of the roof will collapse and fall on his head causing damage.
Now repeat the whole process 2 more times and he is finished and so is this game. Good job!