When you first wake up, you'll be in The Void, so go up the stairs and you'll be paid a visit from The Outsider. After that, you'll actually wake up inside of some container, where you're held captive by Daud. When you gain control, you will want to pick up the bricks on the ground and throw them at the wooden cover above you. Once all of the wood is broken, Blink out of the container. Grab the blade and other items from the nearby table, as you don't have anything at all.
There are two Assassins at the top of the stairs, and it may be difficult to kill them without being noticed, so just rush up and try to take them out without losing too much health. After they're disposed of, jump into the water and swim to the wooden ramp to get onto some dry land.
Keep going forward until you see a chain hanging, so climb it and loot the bodies on the walkway. If you want your gear back, which is most likely, then you need to Blink onto the roof opposite the opening leading to Daud's base. Drop down inside, looting more bodies (getting a gun), then go through the door to Greaves Refinery (nothing at the bottom except for River Krusts).
(OPTIONAL) GETTING YOUR GEAR BACK
Grab some more loot from the dead bodies and go over the fence. There is a Weeper walking around, as well as two more underneath the awning, so stick to the high ground. Go over the gate and on top of the building. To continue any farther, you will need to drop the stairs that lead inside the next building.
In order to drop the stairs, though, you will need to find a whale oil tank to provide power to the switch that drops them. Find the stairs leading inside the building, and you'll find an empty tank dispenser below it. Be very careful here, as there are River Krusts nearby, so don't tread too far as to catch their attention.
Once you have the empty tank, go to the third refill station and fill the tank back up. Take the stairs from before and go inside the building, placing the filled tank in the receptacle and using the switch to lower the stairs. Go up the stairs and follow the path until you can climb a chain, then stop when you hear two Assassins talking.
When the Assassins stop talking and teleport away, Blink up onto the balcony they were on and go inside the building. The drawbridge is up, and you will once again need to find an empty tank, fill it up and place it in the receptacle. The empty tank is behind a wall, to the right of the receptacle; you will need to break the wood before you can go inside. Grab the tank, fill it up nearby and place it in the receptacle.
Once the drawbridge is lowered, use it and climb the chain down. There are two Weepers right away, one by the fire behind you, and another walking around on the walkway. Kill them both and head down the stairs. Drop down to the bottom, pick up one of the whale oil tanks and when you have a clear shot, throw it at the group of Weepers in the middle. Clean up any other Weepers around, grab your gear and then exit the Greaves Refinery.
When you get back to the bridge connecting to Daud's base, Blink to the roof above the part where you would drop and look around the area ahead of you. There are four Assassins in the area, 2 on the ground and 2 on the rooftops. There is a balcony that goes into a room where none of them will be able to see you, so go there.
One of the Assassins is on the roof to the left of the doorway that leads to the next area, and he will teleport to the roof just below that one. When he does that, Blink behind him and kill him. You can kill the other Assassins in the area if you want; when you're done, go through the door into the next area.
In this area, you will see a long wooden walkway over some water. Go a little bit forward, then drop in the water, making sure you go up the nearby stairs. Go forward into the open area and climb up into the broken building. Your point of entry into the big building is through a window, and you can make a distraction by moving up until an Assassin comes up and talks to another. There are a good number of them in the area, so if you pick a fight, be prepared for some action.
Once you make it through the window, there will be two Assassins immediately to your left. Go onto the right side of the area and wait until you get a chance to assassinate them. Go into the next area, which is brightly lit, to find two more Assassins, so kill them and then go out the window by the light.
When you're in the outside area, look up and to the right to spot another window you can Blink to. Move forward and an Assassin will appear and go through the double doors, so be quick and assassinate him. There is another one in the room to the left of the main room, so kill him before going into Daud's office.
This can be a little tricky to do stealthily, as there is an Assassin in the same room as Daud. There is a bookcase on the left wall that you can use to hide, but only if you hug the bookcase and wall at the same time. Occasionally, Daud will stop and read something for a bit, which is your chance to kill the Assassin when his back is turned. It's also your chance to sneak up on Daud and kill him.
Once you kill Daud, grab his key and go all the way back to where you first entered Daud's base through the window. If you didn't notice, there was a pair of doors to the right of the window, which was locked. Of course, Daud's Key will open these doors, which leads to a chain that goes all the way to the bottom, where you'll find a door to the Gateward Tunnel.
In this new area, go to your left and when you spot an opening in the debris on your left, Blink up and get onto the balcony. There is a path between this building and the adjacent one, through the windows that will lead all the way to the rooftops.
Go to the opposite end of the rooftops and Blink onto the metal archway/pole thing. Wait until a train arrives to dump some bodies, which is when you will Blink onto it and ride it back where it came from. Now, you have to get Blink ready, as the train will pass through a Wall of Light, but you can Blink to the platform just outside the Wall of Light.
This will conveniently place you at the whale oil tank powering it, so remove it and make sure you don't touch the tracks, as they are electrified. You want to Blink to the platform on the other side of the tracks, which might make you touch the tracks slightly. If you fall, hurry up and go through the door, as there is a Tallboy standing guard there. Go through the marked door to get to the next area.
TO THE HOUND PITS
Depending on whether you did the side missions for Granny Rags and Slackjaw in High Overseer Campbell and House of Pleasure or not, will determine whether you have to do something extra in the Old Port District area. If you did, check out the Master Sewer Key section for more details. If you didn't, then you can go through the gate and into the next section.
Once you past through the gate, you will notice that there are a lot of Weepers around, so kill them all and then continue. The rest of the way doesn't have much more, although you will need to watch out for the River Krusts. At the seemingly first dead end, the path to continue forward is on the left. Other than that, the path to the exit is pretty straightforward, so once you reach it, use it to finish the mission and automatically start the next.