Most of the achievements are automatically granted as you finish levels. Only one is missable and requires solving a level in a certain way.
You get this after you finish level 2, defeating the Blue and Yellow bosses.
You get this after you finish level 6, defeating the Red boss.
You get this after you finish level 10, defeating the Purple boss.
You get this after you finish level 14, defeating the Green boss.
You get this after you finish level 18, the first where you have to exit the room with each color.
You get this after you finish level 30, the maze with 4 color boxes.
This is missable, you get this in level 34 if you destroy both red walls at the same time.
Like this:
You get this after you finish level 40, defeating the Black Magician boss.