This trophy is obtained for beating all 20 levels in the game.
You will pass through the following zones and face Robotnik at the end of each zone. Number of levels in brackets.
Emerald Hill Zone (2)
Chemical Plant (2)
Aquatic Ruin (2)
Casino Night (2)
Hill Top (2)
Mystic Cave (2)
Oil Ocean (2)
Sky Chase (1)
Wing Fortress (1)
Death Egg (1)
Notes: In Sky Chase zone, you cannot fall. Tails will catch you wherever you jump.
In Wing Fortress zone, the boss's weakness is you can only hit it when the laser is about to be fired, you have to hit where the laser is fired from.
There are no rings or star posts in Death Egg zone and you will face 2 bosses in this stage, so you will need to do the entire stage without getting hit. You can only hit Metal Sonic on the front when he is standing still or skidding.
Big Robotnik can only be hit on the upper chest and only when he is knelt down which is after attempting to land on you and failing.
You can combine this with beating the game in under an hour.