Successfully complete the indicated task to unlock the corresponding character:

Ammand The Corsair: Successfully complete the Pirate Poker sub-mission in the Shipwreck City level.

Black Bart: Successfully complete the Combat Challenge in the Maelstrom level.
Bootstrap Bill Turner: Successfully complete the Defeat Enemies sub-mission in the Prison Fortress level.

Cannon Arm: Successfully complete the Combat Challenge in the Pearl vs. Kraken level.
Capitain Chevalle: Successfully complete the Port Royal level.
Captain Barbossa: Successfully complete the Shipwreck City level.
Captain Elizabeth Swann: Successfully complete the Combo Challenge in the Davy Jones Locker level.
Captain Jack Sparrow: Successfully complete the Prison Fortress level.
Captain Sao Feng: Successfully complete the Singapore level.
Captain Teauge: Successfully complete the Combat Challenge in the Prison Fortress level.
Cotton: Successfully complete the Rescue sub-mission in the Isla de Pelegostos level.

Davy Jones: Successfully complete Pearl vs. Dutchman level.

Elizabeth Swann: Successfully complete the Tortuga level.

Gentlemen Jocard: Successfully complete the Davy Jones Locker level.
Gibbs: Successfully complete the Combat Challenge in the Port Royal level.
Governor Swann: Successfully complete the Combat Challenge in the Pearl vs. Dutchman level.

James Norrington: Successfully complete the Combat Challenge in the Tortuga level.

Lian: Successfully complete the Combat Challenge in the Singapore level.
Lord Cuttler Beckett: Successfully complete the Maelstrom level.

Mercer: Successfully complete the Hit-Combo sub-mission in the Singapore level.
Mistress Ching: Successfully complete the Jackanism in the Tortuga level.
Mordillah: Successfully complete the Combo Challenge in the Prison Fortress level.

Pintel: Successfully complete the Isla Cruces level.

Ragetti: Successfully complete the Combat Challenge in the Isla Cruces level.

Sri Sumba Jee: Successfully complete the Combat Challenge in the Shipwreck City level.

Tai Huang: Successfully complete the Sea Battle level.
Tia Dalma: Successfully complete the Missing Person sub-mission in the Port Royal level.

Villanueva: Successfully complete the Combo Challenge in the Tortuga level.

Will Turner: Successfully complete the Isla de Pelegostos level.

Note: The following swords are listed by order of strength. Successfully complete the indicated task to unlock the corresponding sword:

Cutlass: Default sword
Dutchman sword: Successfully complete the Pearl Vs Dutchman level.
Norrington sword: Successfully complete the Isla Cruces level.
Bone sword: Successfully complete the Davy Jones' Locker level.
Gun sword: Successfully complete the Sea Battle level.
Pirate King's sword: Successfully complete the Shipwreck City level.
Davy Jones' sword: Successfully complete the Maelstrom level.
Ghost sword: Successfully complete the Hit Combo sub-mission on the Isla Cruces level.
Legend sword: Successfully complete the Hit Combo sub-mission on the Maelstrom level.

Gold Idol location in Isla Cruces level - When you get to the waterfall where enemies attack you, defeat them all. Then, walk into the waterfall. Behind the waterfall is a secret room with the Gold Idol.