To obtain this trophy, you need to play through one complete level in one session. Any level does, that is no specific factor. Neither does the difficulty setting.
The trophy will pop when the levels boss is beaten. However it is noteable that you don't need to play an entire level. For example, if you start at the "Main Stage" checkpoint of Level 3, all you need to do is to defeat the boss Todd and the trophy will be unlocked.
You can either join a random game or open up your own.