At the main menu, select the "Community And Extras" option, highlight the "Downloadable Content" selection, and press Triangle. Then, enter "SPTZ17" as a code to unlock the special Spetsnaz Battalion.

Successfully complete one of the following tasks to get a trophy:

Communication Officer (Bronze): Complete the Voice Command Trainer.
First Command (Bronze): Win the Warning Signs mission.
Combat Chain Master (Bronze): Defeat Tanks with Gunships, Gunships with Transports and Transports with Tanks.
Veteran (Bronze): Win one match in each game mode (Conquest, Assault, Siege, Raid).
Road to War (Bronze): Use Secondary Attack for the first time.
World in Flames (Bronze): Win 15 missions in the Solo Campaign.
Weapon Master (Bronze): Win a mission by deploying infantry units only.
Hardened by War (Bronze): Play 20 Single Player Skirmish matches.
No Stone Unturned (Bronze): Win 10 2 vs. 2 multiplayer Skirmish matches.
Call of Booty (Bronze): Win 3 missions in a row in Theatre of War.
Lightning War (Bronze): Win 30 missions without killing a unit in Theatre of War.
Blitz (Bronze): Play 30 missions in Theatre of War city maps.
Tour of Duty (Bronze): Disarm 10 minefields in Theatre of War.
True Patriot (Bronze): Have one unit fight and survive for 10 missions in Theatre of War.
Training Simulator (Bronze): Win 5 missions in a row on the same territory in Theatre of War.
Together We Stand (Bronze): Win a Conquest by securing all Uplinks on a map in Theatre of War.
Always Attack (Bronze): Win 40 missions as Attacker in Theatre of War.
Role of Honor (Bronze): Win a Theatre of War Phase as each Faction; you must have played as that Faction in 10 missions.
Urban Operations (Bronze): Intercept 5 incoming Air Strikes with Electronic Warfare.
Breach and Clear (Bronze): Win 5 missions in a row without losing health in Theatre of War.
Survivor (Bronze): Destroy 100 Critical Buildings in Theatre of War.
BOOM (Bronze): Win a mission within 5 minutes in Theatre of War.
Mass Murderer (Bronze): Play 100 2 vs. 2 Theatre of War missions.
Spearhead (Bronze): Kill 10 units with the Command Vehicle's sentry units.
Strategist (Bronze): Win a match after the enemy has triggered DEFCON in Conquest.
Point of Attack (Bronze): Upgrade 100 Uplinks in any game mode.
Dog of War (Bronze): Kill 15 enemy Command Vehicles.
War is Heck (Bronze): No Uplinks recaptured by hostiles for 10 consecutive Conquest victories.
Destroyer (Bronze): Retake a Critical Uplink within the 3-minute timer in a Siege match.
Command from the Front (Bronze): Win a Conquest mission by Annihilation after the DEFEAT counter was under 30 seconds.
Turning Point (Bronze): Win a Skirmish mission and lose no health.
Field Intelligence (Bronze): Kill 10 enemy Evacuation Helicopters.
Assassin (Bronze): Kill 30 enemy regular units.
Total Warrior (Bronze): Win 20 1 vs. 1 matches in Theatre of War.
Adrenaline (Bronze): Shake to SitRep 88 times.
Sharpshooter (Bronze): Browse your teams either right or left 88 times.
Predator (Bronze): Defeat 6 hostile units with a single unit.
Do or Die (Bronze): Win a Conquest mission by Annihilation after the DEFEAT counter was under 30 seconds.
Angel of Death (Bronze): Kill 10 enemy Evacuation Helicopters.
Cleaner (Bronze): Kill 30 enemy regular units.
Shake that thing (Bronze): Shake to SitRep 88 times.
Team Spirit (Bronze): Browse your teams either right or left 88 times.
Impregnable (Silver): Win Finishing Move without any units being defeated.
Ready for Theatre of War (Silver): Deploy 20 minefields which successfully damage enemies in Theatre of War.
Fast and Furious (Silver): Kill an enemy troop without being spotted.
Total War (Silver): Defeat 6 hostile units with a single unit.
Command & Control (Silver): Achieve a 100% Command Rating in one mission in Theatre of War.
Untouchable (Silver): Win a Skirmish mission and lose no health.
Army of One (Silver): Win 20 1 vs. 1 matches in Theatre of War.
Napoleon (Silver): Achieve a 100% Command Rating in one mission in Theatre of War.
The War to End all Wars (Unknown): Armchair Napoleon.

Additionally, there are two secret trophies:

OpFor (Bronze): Finish "World War III" with all 3 Factions on Hardcore.
Air Defense (Bronze): Intercept 5 incoming Air Strikes with Electronic Warfare.