The multiplayer portion of GTA V, dubbed Grand Theft Auto Online, is now live on Xbox Live. PSN should be getting it shortly.
Update: GTA Online is also now available on PSN.
The update is 59MB and GTA Online comes via a title update in the PlayStation Network and Xbox Live.
After you install that update, you will unlock GTA Online in the Grand Theft Auto V game menu. Then, you can enter the game with the fourth (bottom) slot in the GTA V character wheel or via an online tab when GTA V is paused.
Here's a first look at Grand Theft Auto Online: