Start

You begin with a cutscene where you present the letter to foolish Navy Guards reminiscent of the POC movies. Eventually they get to arguing and you and your crew sneak away to board the Black Pearl. You may hit the ESC key to skip the cutscene.
Opening scene
Opening scene
Goliath
Goliath

Gain control

Once everyone is done viewing it, the battle begins in earnest. You appear on the deck of the Black Pearl. A number (how many?) of level 38 Officers defend the boat. Your best bet for combatting them is the Voodoo Staff with a crew or Voodoo if you're solo.

Take the Wheel

One of your crew must take the wheel of the Black Pearl

Sink the Frigates

Four Navy Frigates will attempt to stop you. Sink them all to continue.

Destroy the Drawbridges

Next you need to destroy the fortifications by the bridges. There are 4 sets total. Aim for fortifications on either end of the bridge. You'll see a health bar above each.
Drawbridges
Drawbridges

Sink the Goliath

Before you can break free to the open sea, the Goliath will attempt to stop you. Sink it and the battle is won!
Goliath sunk
Goliath sunk

Visit Jack Sparrow

Now run back to Tortuga to the bar. Jack will not be at his usual spot at the bar but sitting by Joshamee Gibbs. Talk to Jack and he'll praise you. You will then unlock the Leadership Sailing skill.
Congratulations!
Congratulations!

Now you wait for the next quest to be added to the game.

Referenced by 2 pages pages that link to “Boss Battle (recover the Black Pearl)” — its place in the wiki