There are 12 Trophies that can be collected.

Bronze

Doc's Legacy
Got Doc's Notebook back from Biff

Hello, Young Friend
Talked Emmett into completing the Rocket Drill

A Plausible Explanation
Discovered the origin of the 'new' DeLorean

Fuelmaster
Flawlessly mixed up a batch of Emmett's Rocket Fuel

Fate in flux
Heard about the curious fate of Marshall Strickland

Hi Bob!
Namechecked BTTF Creators

Deja Marty
Perfectly repeated dialog while repeating Doc's temporal experiment

Silver

Back in Time
Traveled back in time to rescue Doc

Grandpa Got Served
Delivered the subpoeana to Arthur McFly

Junior Bootlegger Badge
Managed to get 190-proof alcohol delivered to Emmett's

Jailbreak!
Acquired Emmett's Rocket Drill

Gold

A Bicycle Built for Two
Rescued Doc from Kid Tannen

http://www.ps3trophies.org/game/back-to-the-future-episode-1/trophies/