This quest is unlocked after completing "The Fireship" quest. To start it, you must travel to North Carolina by sailing to the location marked on the eastern side of the map. Two enemy ships will be waiting for you there, which you must destroy first.
Edward sails to North Carolina to convince Blackbeard to return to the West Indies.
Follow the suspicious pirate who catches Edward's attention at the end of the opening cutscene. This pirate will head inland. Climb up the boards and scaffolding onto the rocks on the left side to follow him, and you will see the pirate meet with the British soldiers.
Note: You can get past single soldiers by sneaking past them quietly - but why risk being detected? :-)
Hang the soldier who takes up watch near the palm leaf pile - the second soldier standing on the other side of the path will not notice. Leap down into the leaf pile, move into the bushes, and follow the pirate and the captain through the hole in the fence.
Continue following the captain and the pirate. When they reach the building and turn right, head straight into the building instead.
Wait for them to pass the soldier standing next to the crate with his back to you, then eliminate him quietly.
Then, head straight along the crates and barrels to the end, turn right, and keep going forward. Once the captain and pirate pass the next soldier, approach him and assassinate him.
Climb up the wagon leaning against the wall - a few steps further you should see two soldiers you can kill using Double Assassination.
Simply run up the stairs in front of you - once you reach the top, you will see the traitorous pirate fire a flare, which will start the enemy ships' bombardment of the coast.
You must now reach the deck of your ship and take control of it. Jump down and move forward while avoiding fire damage. You can cover the entire distance by jumping across rooftops - beams - and similar structures, or you can cover part of the distance by swimming on or under the water.
The battle will take place in a relatively small area - and not only you and the British man-of-war will be involved, but also other British and pirate ships. First of all, don't get yourself killed. Use all the means at your disposal to drain the British ship's health bar and incapacitate it. Since it's a fairly large vessel, mortar fire seems to be effective. Of course, this works both ways, so watch out for the amber spots appearing on the water near you as well.
Once the man-of-war is incapacitated, sail up to it and board the ship.
Climb aboard the enemy vessel and begin killing British soldiers - fighting them, shooting at them, or detonating gunpowder kegs. After killing a required number of them, a cutscene will trigger - a certain era in the history of Caribbean piracy will come to an end.
After the cutscene, you will end up in the water - luckily, the Jackdaw is right next to you, so simply climb aboard and take the wheel.
Sail straight ahead toward the marker - avoiding any ships you encounter along the way. Once you sail far enough, another cutscene will begin.
And that is the end of this quest. The next main quest will be unlocked - Vainglorious Bastards - and you will receive a new treasure map - Blackbeard's Treasure Map.