Accomplish the indicated achievement to get the corresponding number of Gamerscore points:

MobFace (5 points): Create a mobster using MobFace.
Getting Made (5 points): Recruit your first soldier.
First Crime Ring (15 points): Control one crime ring.
Fortified Venue (5 points): Max out all guards at a venue.
It's Not Personal (20 points): Eliminate a rival family by successfully bombing their compound.
Contract Killer (15 points): Eliminate 5 made men with the appropriate kill condition.
Lockpicker (5 points): Crack five safes.
Enforcer (15 points): Complete 10 execution styles.
Bank Job (5 points): Heist your first bank.
Let Me Upgrade You (5 points): Upgrade one of your made men's skills or weapons.
Three Crime Rings (40 points): Control three crime rings.
Five Crime Rings (50 points): Control five crime rings.
Organized Crime (75 points): Control all crime rings.
Shakedown (25 points): Control all extortable businesses.
Executions In Style (75 points): Complete all execution styles.
Bag Man (20 points): Heist each bank once.
Safecracker (75 points): Crack all safes.
Torch The Joint (5 points): Send your men to bomb a venue from the Don's View.
Pulling The Strings (5 points): Send your men to attack a venue from the Don's View.
25 Massacred (10 points): Kill 25 mobsters.
100 Whacked (15 points): Kill 100 mobsters.
250 Iced (20 points): Kill 250 mobsters.
500 Empty Suits (25 points): Kill 500 mobsters.
750 Sleeping With The Fishes (30 points): Kill 750 mobsters.
BlackHand Brutality (10 points): Kill 25 mobsters only using BlackHand attacks.
Full Of Lead (10 points): Kill 150 mobsters only using firearms.
Gun Smuggler (10 points): Find a level 2 firearm upgrade.
Modified Firepower (20 points): Find a level 3 firearm upgrade.
Right Hand Man (10 points): Promote one of your made men to Capo.
Second In Command (15 points): Promote one of your made men to Underboss.
This Thing Of Ours (25 points): Recruit a full family tree.
Accept This As A Gift (30 points): Earn one of each favor.
Welcome To The Gun Show (30 points): Collect all level 2 and 3 firearm upgrades.

Additionally, there are eight secret achievements:

Secret achievement (25 points): Unknown.
Secret achievement (25 points): Unknown.
Secret achievement (20 points): Unknown.
Secret achievement (30 points): Unknown.
Secret achievement (40 points): Unknown.
Secret achievement (50 points): Unknown.
Secret achievement (10 points): Unknown.
Secret achievement (75 points): Unknown.