When you get back to the Hound's Pit pub, collect everything you can in the area that you haven't picked up yet and speak with Piero to buy more items. Avoiding any major spoilers, we'll just say that this will be the last time you return to the Hound's Pit in a peaceful setting, so you should make the most of your time here now.
There are three optional objectives from the last mission that can earn you Runes now if you speak to the right people. If you dueled Lord Shaw, speak with Pendleton to get a Rune. If you spared the Boyle Sisters and instead gave them over to Lord Brisby, a note at the entrance to the pub will notify you that a Rune has been delivered. Finally, Callista in the tavern will ask you to go find Emily. She will be either in Corvo's room, at the base of the tower where you met Pendleton earlier, or near the entrance to teh Abandoned Apartments area. Use Dark Vision to find her, and she will give you a Rune.
Now that you're ready to start up the mission, head over to Sam's boat and tell him that you want to complete your final task. The map here is the same as it was in the prologue, except bigger and infested with enemies. You will need to get up to the tower's base by jumping and Blinking upwards; a beachhead also leads up to the tower, but it's infested with enemies crawling around everywhere. Instead, use the rocks to the left to get up and over the patrols. In the waterlock, there is a Bone Charm in one of the side ducts. Use an upgraded Blink to reach it, and crawl along the platform to the side to nab it.
Once you reach the top of the area, power down or sneak around the Wall of Light by either taking away the whale oil fuel cell from the generator or sneaking underneath of the bridge where the tallboy patrols. Use the moat below to get under it without being found out.
As you pass the bridge, you will enter the area right outside of the tower, which just so happens to be heavily infested with enemies. This situation seems bad, and indeed it can easily become more than most can manage if you let the enemies discover you. Armed City Watchmen patrol the area directly outside of the twoer's base, while tallboys roam the sidewalk path in front of it. A watchtower sits to the side of the tower itself, prepared to fire rockets down upon anyone dumb enough to get close. Unfortunately for you, you will have to get close if you want to have any hope of getting inside.
If you have a level 2 Possession skill, this section becomes a lot easier. Simply possess one of the City Watchmen at the front gate and head down the side path to the right to avoid detection. You can also use Blink to try and get past the section, using the battlements above to keep away from the guards below. Use Blink between the platforms above to slip through a little hole in the front wall. You are now in the Keep's front yard. There's a Rune here, in the bunkhouse on the side of the entrance with the turret watchtower. Deactivate the tower first unless you plan on getting spotted and bringing a premature end to your mission. Time your approach so that when the tallboy moves, you are concealed. The Rune can be found under one of the bunks inside of the watchtower.