This quest unlocks after completing the "Devil's Advocate" quest. To start it, sail to the north of the map where you will find Stede Bonnet's ship. After talking to him, you will automatically fast-travel to Charles-Towne where Blackbeard will already be waiting for you.
Edward offers to help Blackbeard acquire medicines from the stubborn residents of Charles-Towne.
Hint:
The path is linear, and your top speed is capped. Head forward along the river and stick to the center of its bed - after a moment, you will reach an outpost and have to step ashore for a brief moment.
Once you step ashore, move across the rooftops toward the location of the alarm bell. It is guarded by five guards, two of whom are patrolling. Start by jumping into the hay cart and killing the soldier standing next to it.
Next, sneak behind the pair of soldiers standing slightly in the back - keeping to cover behind the crates - and execute a Double Assassination. Finish off the remaining two soldiers and sabotage the alarm bell.
Hint:
Return to the Jackdaw - which will sail a bit closer - and take the wheel again. Since you killed all the guards, no one will spot you.
Continue down the river until the gunboat moors and the water becomes too shallow for the Jackdaw, so you will pull up to the bank and step ashore again.
Walk a short distance on land and you will see that the soldiers have switched to a rowboat.
Move forward and go through the shipwreck, then jump to the other side of the river and hide in the tall grass on the opposite bank.
Climb the wooden ramp to the upper floor of the ruined building, cross to the other side, and then use the stone columns and wall fragments to reach the next building.
Continue forward by jumping across the chandeliers, over a wooden beam, and through the hole in the wall then drop down. You will witness a crocodile kill one of the soldiers. You should be able to easily bypass two guards by going to the right, but you can also double assassinate them.
Climb up another wall, from where you will see that Officer Ryder has stepped ashore. As you approach the buildings, the officer will head further inland, and shortly after, a cutscene will trigger showing you who holds the key to the medicine storehouse.
After the cutscene, the captain will break into a run - you can sprint after him as fast as possible, but you don't have to. There is no time limit - so it is best to kill the first two soldiers who stand in your way, one of whom is the captain, heal yourself if necessary, and only then continue on your way.
Climb onto the roofs of the huts and run across them, then jump across the beams between two buildings. A bit further ahead, slide under the collapsed beams.
Don't worry about the group of soldiers charging at you - a massive explosion will take care of them, allowing you to jump ahead across the remaining beams of the walkway.
You can daze the group of soldiers guarding the gate by throwing a smoke bomb - then quickly run up the fortifications using the cart leaning against the wall to the left of the gate.
On the other side is a small pond with stakes driven into it - you can jump across them to proceed, but watch out for the crocodile lurking there. Alternatively, you can drop down earlier and run across the bridge.
You can get past the next group of soldiers by jumping over the fence - scaling a few crates to the right of the gate.
You are getting close to your destination - get past a few more soldiers,...
..and shortly after, you will reach the barn where the captain runs inside. The soldiers will close the gate behind him - so you must climb the crates to the left of the gate and get inside through the upper opening.
Finish off the captain - ideally with a well-aimed headshot.
This will complete the quest, automatically returning you to the tavern in Nassau where the next main quest will be unlocked - We Demand A Parley.