To get this, you must beat the dark world version of the world 6 boss. There are two requirements to be able to do this:
-You must have defeated the light world version of the world 6 boss.
-You must have completed at least 85 dark world stages.
Dark world stages are unlocked by clearing the light world version of that stage within par time (A+ rank). You can do any 85 stages from worlds 1-7, as order doesn't matter, but I'd recommend starting with every dark stage in worlds 1-4, which will give you 80/85. From there, do any stages you can from the last three worlds. If you're having trouble in world 5, 6-1X and 7-12X are relatively painless. The dark world boss marker in world 6 tells you how many you have done.
The boss fight in world 6 is mostly the same as it is in the light world. The first part has more saws and a slightly stricter path, but is otherwise identical, and if you've made it this far you shouldn't have too many problems. In the last room, you have to hit the button on the ceiling, then the button on the right to drop Dr. Fetus in the pit.
The second part is much tougher than the light world version, but again, you just need to make it back to the left end of the stage. There's a LOT more saws and a MUCH stricter path than the light world, but again if you've made it this far, it's doable. The achievement pops as soon as the cutscene after this area starts.
Edit: Added video walkthroughs of the last stage, thanks DarkxMaterials!