Stay within range of Lee - 50m
Do not shove anyone during the chase
Do not take any fire damage
Take off after Lee at a run but hug the left-hand side of the dock until the powder keg explodes in front of you. AFter that move over to the right-hand side of the docks and start free-running on up. This dodges 2 Soldiers who formed a Firing Line from taking a shot at you.
After that keep to the right to avoid a second firing line. Now move over to the left after you pass the boxes. A third firing line lays in front of you. Head over to the left and jump at the left-hand corner nearby the line. This will keep you in range of Lee and avoid getting shot.
Turn to the left and follow Lee toward the boat. As you enter the boat the objectives will update and you will gain the "Do not take any fire damage" optional objective.
Head inside, slide through the first narrow opening and then break over to the right. Climb on up and turn to the left. Climb on up and make your way over to the left. Duck through the not burning parts and start working your way quickly across the far side of the ship.
When you get to the end of the free-run course you will see a rope to cut. Do so to quickly get carried up to where Lee is approach him to trigger the next part of the chase. Just keep after Lee as he makes toward the deck of the ship. Just run him down to trigger the next cutscene and objective.