Probably the easiest romance in the game, Madison is a very easy girl to court. There aren't that many conversations with Madison, meaning romancing her is easy, but that also means screwing up is as well. When you first start talking to her, stick with professional answers. Eventually, when you get to the mission Investigate Marburg's Villa, you will be given the choice between choosing either Mina or Madison as your handler. For the sake of romancing Madison, choose her. You'll lose one like point with Mina, but that can very easily be made up during the course of the game.
After finishing that mission, you will talk to Madison at your safe house. From here on, use nothing but Suave responses. After finishing the next two missions you receive from the Villa, you will be given the choice to either check on Madison, or leave her be. Check on her, and during the following conversation, use Suave answers as always. Once the scene ends, the achievement will unlock.