There are 48 Trophies that can be collected.

Bronze

Smooth Operator    
Press 10 TARDIS console buttons    

Thorough    
Scan 10 objects    

Head scratcher    
Collect 5 Hats    

Paper trail    
Collect 5 Diary pages    

Brimming with enthusiasm    
Collect 10 Hats    

Short story    
Collect 10 Diary pages

Puzzler    
Complete the game on puzzle medium difficulty   

Silver

Hat's off to you    
Collect 20 Hats    

Long story    
Collect 20 Diary pages    

Flying Doctor    
Complete the game in 300 minutes

High IQ    
Complete the game on puzzle hard

Gold

Diarist    
Collect all the Diary Pages    

Mad Hatter    
Collect all the Hats

Platinum

Time Lord    
Collect all Trophies

34 Secret

Bronze

Crate work    
Use the gold bullion crate to smash your way out of the Bank of England

Fast on your feet    
Beat the Cybermen to the Underground refugee camp    

On the right tracks    
Defeat the Cybermen in the Underground refugee camp    

Stunning work    
Stun 5 Cybermen    

A Plumb Job    
Re-route the gas pipe plans in Victorian London    

Hack Job    
Open the security door in the office block    

Tested to Destruction    
Re-route the thermionic consoles in the Cyberfactory

The Open Road    
Lower the drawbridge on Old London Bridge    

No time to hang around    
Collect the Stasis Field Generator in the Silence Base

Soundly Beaten    
Sonic disarm 5 Silurians    

Hello Sweetie    
Wake up River in the tomb    

Silently Silencing the Silence    
Outwit the Silence without being checkpointed    

Toxic Relationship    
Defeat Vekkis    

Goodbye Blue Boy    
Outwit the Strategist Dalek    

The Eyes have it    
Quick-shoot 5 Daleks

Ye Doctor    
Discover the statue in the Elizabethan quadrangle    

Winch Way Next    
Activate the crane winch in Old St. Pauls    

A steal    
Defeat the Silence and steal their time capsule    

Hello Sexy    
Find the TARDIS in the Dalek time lock    

Sent packing    
Send the Dalek Supreme to Antarctica

One in the eye for the good guy    
Shoot the Emperor Dalek in the eyestalk    

Geronimo!    
Defeat the Emperor of the Daleks    

Clever clogs    
Complete Data Link puzzle in 60 seconds    

Smart Cookie    
Complete Re-router puzzle in 60 seconds    

Clear Vision    
Complete Mind Map puzzle on Medium or Hard in 60 seconds

Silver

The Stormcage Redemption    
The Stormcage Redemption    

Pucker Up    
Kiss 6 Stormcage Guards    

Quick Getaway    
Escape Stormcage in 8 minutes

Deleted    
Finish the Cyberfactory in 20 minutes

Elementary    
Finish return to Victorian London in 15 minutes

Beg for Mercy    
Power-shoot 5 Daleks

Exterminated    
Finish time-locked London in 20 minutes

Timey Wimey    
Complete a Time Corridor in 2 minutes    

Universal Destruction Avoided    
Complete the game

http://www.ps3trophies.org/game/doctor-who-the-eternity-clock/trophies/