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

Enforcer (Bronze): Complete 10 execution styles. 
Contract Killer (Bronze): Eliminate 5 made men with the appropriate kill 
condition. 
Three Crime Rings (Bronze): Control three crime rings. 
Five Crime Rings (Bronze): Control five crime rings. 
Lockpicker (Bronze): Crack five safes. 
MobFace (Bronze): Create a mobster using MobFace. 
Let Me Upgrade You (Bronze): Upgrade one of your made men's skills or weapons. 
Torch The Joint (Bronze): Send your men to bomb a venue from the Don's View. 
Pulling The Strings (Bronze): Send your men to attack a venue from the Don's 
View. 
Gun Smuggler (Bronze): Find a level 2 firearm upgrade. 
Modified Firepower (Bronze): Find a level 3 firearm upgrade. 
Bank Job (Bronze): Heist your first bank. 
100 Whacked (Bronze): Kill 100 mobsters. 
250 Iced (Bronze): Kill 250 mobsters. 
25 Massacred (Bronze): Kill 25 mobsters. 
BlackHand Brutality (Bronze): Kill 25 mobsters only using BlackHand attacks. 
500 Empty Suits (Bronze): Kill 500 Mobsters. 
Full Of Lead (Bronze): Kill 150 mobsters only using firearms. 
Fortified Venue (Bronze): Max out all guards at a venue. 
Right Hand Man (Bronze): Promote one of your made men to Capo. 
Second In Command (Bronze): Promote one of your men to Underboss. 
Getting Made (Bronze): Recruit your first soldier. 
Shakedown (Silver): Control all extortable businesses. 
It's Not Personal (Silver): Eliminate a rival family by successfully bombing 
their compound. 
Bag Man (Silver): Heist each bank once. 
750 Sleeping With The Fishes (Silver): Kill 750 mobsters. 
This Thing Of Ours (Silver): Recruit a full family tree. 
First Crime Ring (Silver): Control one crime ring. 
Accept This As A Gift (Silver): Earn one of each favor. 
Welcome To The Gun Show (Silver): Collect all level 2 and level 3 firearm 
upgrades. 
Executions In Style (Gold): Complete all execution styles. 
Organized Crime (Gold): Control all crime rings. 
Safecracker (Gold): Crack all safes. 
Platinum Trophy (Platinum): Act like a mobster, think like a Don.