BattleBlock Theater: Theater Critic Achievement Guide

1. Local Game
   
2. Furbottom’s Features
   
3. Feature
   
4. Normal Mode (Or insane, if you’re insane)
   
5. Customize your character and ready up
   
6. Complete all levels (except encore)
   
7. Achievement should pop after the boss levels
   

You do not need to actually rate any levels, as other guides may indicate. That feature was removed when the servers was taken down in 2018.
And no, you do not need to rate levels in the Steam Workshop. I rated 10+ levels without the achievement popping.