With the "Benjamin" run aground, Edward need to head onto the island and deal with Hornigold. Disembark via the rope and swim to shore. You are not under any stealth requirements here and can fight your way almost completely to Hornigold. It is best not to bring the fight that close to him as it will ruin your chance to Air Assassinate him.
Once on the beach closest to the Jackdaw duck into the nearby Stalking Zone. Deal with the Soldier who is standing nearby. Now cut directly across the shallows in front of you, toward the ruined temple on the rocks. If a patrol of 2 Soldiers is in front of you, kill them.
Turn to the left when you make it to the cliff face. Make for the Eastern heights, killing off the Soldiers in your way. Climb up the Western side of it to get to the higher area. Sweep East and take out the Guards who are in your way. With a little luck, they're all looking the other way and you can just slaughter your way on through. Head back to where you climbed up and duck toward the covered walkway. Use the outside of it to climb up to the next level of the island. Head to the South from there, toward the tree that hangs over the gap between the Eastern and the Western heights of the island. Get to the tree and use that Free run course to get over to the Western island.
Once on the Western island, move to the Southern edge of it and slip along the very edge. Follow it to the Western side and then start to the North toward the Southwestern side of the temple ruins. Just start climbing on up at the closest point. Climb on up toward the Viewpoint there and you will have a prime opportunity for an air assassination on Hornigold. Just move along the top wall and get close to him to ensure you are in range to get him.