Go up the stairs at the beginning and use the chain to get to the next section. There are four guards here, with the easiest one to kill being the one who walks the length of the area. The next guard you should go after is the one inside the structure, then the one who walks towards the door you need to use.
Go through the door and go upstairs until you come to a cart and a desk, which has a key on it. You can either use the key on the door on the floor below you, or you can use the cart to get across the area. Of course, you need to power up the track, which can be done by getting a whale oil tank on the level above the cart and placing it in the receptacle. Note that if you use the cart, you must ride inside it and in stealth mode to avoid detection.
Whichever method you choose, you'll find around 4 guards in the area. If you used the cart, you can bypass the guards by Blinking to the door to the next area. If you used the door, pick off the guards when you get the chance, kill them and proceed into the next area.
DRAWBRIDGE WAY
When you first enter the area, you'll see a guard talking to someone named Pratchett. After they're done talking, the guard will go down the stairs and Pratchett will go inside his house; both are easy pickings, so kill them both. To get to the next area, you can go through the streets, where you'll find a Wolfhound, or you can go via the rooftops, which is a better idea.
As you approach the front of the bridge, don't go too far as there is an Arc Pylon, which fires electricity at you and can kill you in two hits. There's a path that leads down to the waters, and you can get across to right near the Arc Pylon. Right before the stairs that lead up to the Arc Pylon, you can find the control panel hidden behind a wall.
If you have a Rewire Tool, use it to make the Arc Pylon attack the guards. With the Arc Pylon on your side, use it for cover and try to lure the other guards within its firing range, while picking off the rest with your pistol and crossbow. Once the coast is clear, you can continue forward.
There's another Arc Pylon on the bridge, before you go up the stairs. If all of the enemies are dead from the previous fight, go down the stairs and find the whale oil tank that powers it, so remove it and go back up. Climb up the long flight of stairs and into the control tower, flipping the switch to make the bridge raise. Blink up to the raised bridge and go over, going up and taking out the oil tanks to disable the spotlights.
With that out of the way, it's time to start your descent. Note that there's an Arc Pylon located in the same spot, on the bottom, as there was on the other side of the bridge. As you go down the stairs, you'll come to a landing with what seems like an extra piece attached on. Look out over the railing and you'll see a guardpost below, so Blink down to it.
This is right on the same level as the Arc Pylon, and the whale oil tank powering it is in a similar position as the one across the bridge. Of course, there are guards here, so kill them if you can before moving on. There is a thug in a jail cell, who wants you to help him escape. This is completely optional and the rewards aren't too good, but if you want to do this short side mission, you're free to do so.
(OPTIONAL) FOLLOW THE SCAVENGER
The key to his cell is located on a table in the same room, so grab it and unlock the cell. He will come out and start walking outside and down the steps, along the pier. Follow him and he will stop before an opening, asking you to take care of some River Krusts. These are tricky little enemies, as they are only vulnerable when they open up to fire projectiles at you. There are around 5 of them, so take them all out.
Once they're all gone, the thug will continue on and up the stairs. He will stop by the gate and mention that the guards have the key. It is located in the one guard structure, closer to the gate, but you need to be careful of guards. Kill any guards in the vicinity, then go and grab the key, but don't go back to the thug just yet.
Go into the building that is nearby, and you will find two more thugs. Kill them both, grab the pearls in the room, and then head back to the thug by the gate. What would normally happen is that when you go and unlock the gate, the thug would use the sink and call his two friends, who will all attack you. If you kill his friends first, you reduce the 3 on 1 to a 1 on 1.
MIDROW SUBSTATION
Whenever you're done with everything in the previous area, use the door to get into the next main area. You'll be in a building when you first enter the area, so go out and onto the rooftop. There are a number of guards in the area, three of which are on the ground and one is on the catwalk above. Follow the vent to the right and you will be able to Blink onto the roof of the guard structure.
Kill the guard walking on this walkway, then flip the switch to stop the one spinning gear. Kill the rest of the guards in the area, then go through the opening caused by the stopped gear. Take out the whale oil tank powering the Wall of Light and continue to the next area.
SOKOLOV'S HOUSE
When you first arrive in the area, go forward and take the alley to the right. Go up the stairs, but not all the way, as you will want to Blink onto the roof opposite the second flight. Go onto the broken platform and you'll see a chain nearby. Climbing the chain will put you right outside Sokolov's room, but be careful of the guard walking around nearby. Note that you will fail the mission if you kill Sokolov, so go inside and sneak behind him, choking him out.
If you decide to go through the front door, you will encounter two guards at the entrance. When you go inside, there will be a number of guards on the bottom floor (as well as the only Sokolov Painting in this mission), one of which is an Overseer with a machine that renders your powers useless.
After they're all dead, go to the isolated Wall of Light and find a Rewire Tool on a table, as you'll be needing it. Go up the stairs and you'll see a Wall of Light, which is what you will encounter on the other path. The only way to get through them is to find the control panel in a room right next to the one Wall of Light and use a Rewire Tool on it; doing this will allow you to pass.
Go up the stairs in the room between the Wall of Lights and you'll find another two guards in the hallway. The one door is for the elevator, which is not functional, as it needs a whale oil tank. A dispenser and filling machine are in the next room, so you can power the elevator if you want (unless you kill everyone, which wouldn't matter).
In this room are some steps that lead to the top, where you will find a guard and a door that leads to Sokolov. Sneak up behind him and render him unconscious, then pick up his body and start heading towards where Samuel is located. Once you get to his boat, you can head back and the mission will be finished.
