Romancing Mina isn't too difficult to do, but it is a long process, as you won't get this achievement towards the end of the game. When you first meet Mina, stick with Professional responses in order to earn her respect, with the occasional joke or two. As you get her like up more and more, you can than start to use Suave responses more and more, and she will reciprocate.
Make sure that during any missions that include either CIA agents, Marines, civilians, or local law enforcement, that you do not kill them. Knocking them out is okay, but if you kill any of the groups mentioned above, you will anger Mina, losing one like point per kill.
Eventually, when you finally finish Moscow, Rome, and Taipei, Mina will meet up with you. Stick with Suave responses here, and when you're finally given the option, kiss her. From then on, watch the scene for the achievement.