As with the Gentleman Caller achievement, some prerequisites must be met. You must complete both Granny Rags side missions in Mission 2, and you must complete the Slackjaw side mission
in Mission 3. Later, in Mission 7, when in the tunneled sewers just before the end of the level, a thug will be laying on the right side of the tunnel injured. Talk to him, and then enter through the pipe and go through the fairly long section of sewer water and climbing until you see an open area with Granny Rags pacing and Slackjaw chained up. Save before this in order to get both Street Conspiracy and Gentleman Caller.
To help Slackjaw, you must eliminate Granny Rags. Do this by going up the house and burning a piece of her "true self." The item you need to burn just under a pillow on the bed in the same room as the incinerator. Open the incinerator, place the cameo inside, close the incinerator, and kill Granny Rags. She cannot die if you don't destroy the cameo. Once you free Slackjaw and collect your reward, the achievement should pop.
Just talk to Granny Rags and cook Slackjaw by placing him in the tub after killing him with your sword to unlock the Gentleman Caller achievement.
Note: Killing Granny Rags (or Slackjaw) will count as a kill, so achievements like Clean Hands will be voided. You also do not actually need to interact with either Slackjaw or Granny Rags before you place the cameo in the incinerator. I was actually able to do all of this without being "altered" by going straight to the cameo, destroying it, and killing Granny Rags undetected.