Important: In order to get Slackjaw's final side mission, you MUST meet Granny Rags in Mission 2 and complete her missions.
Slackjaw is found in Mission 3: House of Pleasure in his Distillery. It is very early on in the level. He has two side missions for you. Complete them and you won't see him again until late in Mission 7. While in that mission you will find a thug in the sewers on your normal route to escape the area. He will give you a key which you can use. At the end of this new route, you will find Slackjaw and Granny Rags.
WARNING SAVE YOUR GAME
Complete his mission and you will get the achievement. Reload your save and complete Granny's mission instead to gain the related achievement for her side missions.