This quest unlocks after completing the "A Governor No Longer" quest. To start it, you must return to Princípe, which you can do in a similar way as during your first visit to Africa. This time, to sail there, you will first need to get rid of four ships - one brig and 3 schooners.
Hoping to recover the Observatory device, Edward pursues his second of three primary targets: Bartholomew Roberts.
Roberts is located in the upper camp, within an area surrounded by a palisade that has only one entrance. How you get there does not matter - even if you draw aggro - as long as you survive when you reach the spot. This time, there will be no soldiers on the island - only Roberts' pirates.
You start on the eastern side of the island, next to the docks where Burgess's ship was previously anchored during the "Trust Is Earned" main quest.
After coming ashore, you can climb the watchtower located near the beach and kill the marksman on top of it.
Next, leap down onto the roofs of the buildings on the left and follow them a bit to the northwest.
Cross the tree to reach the rooftops on the other side of the street, then climb the scaffolding visible on the right side of the screenshot below.
Once you climb to the top of the scaffolding, follow the path leading uphill. Watch out for the rifleman standing on the low building to the left, as well as the shooter on the tower with the viewpoint. Hide in the tall grass.
Then, take the path to the right leading uphill and hide in the tall grass.
Directly ahead, you should see the gap in the palisade that you need to reach.
When you try to pass through that gate, Roberts will emerge from behind the barrels in front of you, shoot you - dealing a bit of damage and briefly stunning you - before jumping off the cliff onto his ship and sailing away.
The Jackdaw will sail to the spot where Roberts' ship was standing moments ago, so recover from Roberts' shot that knocked you down, quickly head toward the cliff, jump down to get onto your ship's deck, and take the wheel.
The Royal Fortune will start moving away through a narrow passage. Take advantage of this and begin shelling it with your mortar. The shots take a moment to reach the target, so lead your aim slightly ahead of it.
As soon as you see the Royal Fortune drop barrels behind it, destroy them as quickly as possible.
This way - depending on how much you have upgraded your ship - you should deal a significant amount of damage before it reaches open waters.
Once on open waters, other ships hunting Roberts will appear. Focus on depleting Royal Fortune health bar as quickly as possible, and once you do, proceed to board the ship.
You can board the ship, approach Roberts, and defeat him in sword combat - but he is no easy opponent to overcome. A better option is to try killing him with a single assassination strike.
Roberts is at the stern, near the steering wheel. So, board the ship somewhere around its middle section and use the crane to quickly reach the mainmast.
Next, leap across the rigging to reach the mast near the ship's stern.
Wait until Roberts is within your reach and perform an Air Assassination on him. Be careful not to accidentally kill one of the other sailors on deck.
Once Roberts is dead, a cutscene will trigger, after which you will return to the Caribbean. This quest will end, leaving you with the last Templar to kill - Grand Master Torres - Tainted Blood main quest.