The page below of our AC 3 Remastered guide contains a walkthrough for Public Execution, along with the list and walkthrough for optional objectives.
After the cutscene, head forward all the way to the gallows.
During the execution, you have to press the Assassin Guild button when prompted.
Optional objective: Washington's bodyguards must survive.
Optional objective: Kill enemy militia - 2/2.
Optional objective: Complete all optional objectives during a single playthrough.
Soon afterwards you will land on the ground and will have to kill Hickey. Start off by running after your target and when he stops, quickly press the jump button to knock him down.
You should do it before he manages to kill Washington's bodyguards. Unfortunately whether you will make it in time is a matter of luck mostly. If Hickey doesn't get blocked by someone in the crowd, he will easily be able to kill one of the guards. To complete the optional objective, you will have to replay this part until you eventually manage to make it in time.
With Hickey on the ground, stand beside him and wait for the militia to attack. You should counter their attacks and afterwards perform a series of three deadly blows, with the last one lethal for Hickey. If there's only one militia, reload the game or knock over the Templar again in hope that someone approaches you and you will be able to use him for full synchronization.
After Hickey's death, follow Achilles, open the door and end the sequence.