If you complete the task indicated then you will unlock the outfit shown below.

Outfit	Task
Button Man	Complete the Badge Pursuit Challenge. Gives all yours weapons extra ammo.
Chicago Lightning	Sign up for the Rockstar Social Club. Increases your accuracy with shotguns and machine guns.
Golden Boy	Have Cole Phelps get promoted to the Traffic Desk for the LAPD.
Hawkshaw	Reach Rank 18. Allows you to take more damage.
Sunset Strip	Reach Rank 8.
Sword Of Justice	Reach Rank 3.
The Broderick	This was an Amazon pre order bonus. It increases melee damage and resistance to damage.
The Outsider	Reach Rank 13.
The Sharpshooter	This was a pre order bonus with various retailers. Increases your accuracy with pistols and rifles.