This page of the Assassin's Creed III Remastered guide contains a walkthrough for the final mission in the game - Chasing Lee.
To begin the last mission, head to the inn at Boston and interrogate the man inside. He will tell you that Charles Lee can be found in the docks.
Optional objective: Stay within range of Lee - 50m.
Optional objective: Do not shove anyone during the chase.
Optional objective: Complete all optional objectives during a single playthrough.
Use Eagle Vision in the marked area of the shore and find the Templar. As you approach him, the chase will begin.
Start off by quickly jumping to the seashore to avoid the first group of guards.
The second and third group can be avoided by jumping along nearby barrels. You can also run into the soldiers and perform a dodge to make Connor jump forward and avoid getting hit.
The last group can be avoided by cutting the left turn. A couple more steps and the optional objective will be complete. You won't have to worry so much about the distance of shoving pedestrians, though you still won't be able to lose Charles.
Optional objective: Do not take any fire damage.
Start off by turning right to the burning shipyard and slip under the wooden blockade.
From there you have to run onto the wall on the right and afterwards jump left.
By the end of the ship there will be a cinematic after which you will have to jump through a hole in the wooden barricade and run further to the right.
To catch up with your target, you should run around the construction and slide under one of the wooden barricades.
After you reach the target, slowly move towards the shore. A man waiting there will take you to where Lee went.
Enter the tavern and watch a long cinematic. Afterwards you will once again take control of Desmond.
Open the big red gate and afterwards run straight along the path. That way you will finish Assassin's Creed III. Congratulations!
After the credits roll, you will enter the shoes of Connor once again. Now you can complete all the previously ignored additional missions and find the collectibles. Apart from that you have to epilogue missions, in the form of short cutscenes.