This is a surprisingly easy map (especially on the console version).

PC: The only way to escape is to get $250 bucks, and head over to the docks at night, there will be a boat w/ Officer Birdseye X, you then finally get back to the main island, and then you escaped.

Console: On the console version, you do not need to get $250, rather just get out of the prison and head over to the dock. The gate that leads outside has a contraband detector and is locked by a red key. A really simple and easy strategy is to beat up a guard and take the red key (make a mold and then the plastic red key to make it even easier) then go south to the bottom of the prison and exit, then go right, to the dock. If you do this during a Lockdown (preferably during Role Call as most of the guards will be north, near the top of the prison), you will have more time quickly get through the door, scotch-free.

Fun Fact:

The Alcatraz prison is based off of a real life prison in California where three inmates escaped out of their cells and on a raft. However, where or if they landed on shore is unknown. The prison shut down March 21, 1963.