Here's a video to help you all out to unlock this achievement. There are parts where your court approval will go down if you take to long to not be in the ballroom such as going to the library or garden during a few parts of the quest. Make sure to talk to the council members to raise your approval, give Leliana secrets that you encounter during the quest, collect coins and put them in a pool outside in the courtyard and during a sequence in the story, when dancing with the Duchess, choose the middle right option every time, and say you trust the Inquisition (40-50pts can be gained here alone)
^I give credit for that last bit to Floyd XBA over at xboxachievements.org. Enough talking! This achievement is missable as well but the video should guarantee a pretty good idea of what to do. If you have trouble, save often and follow the video and my advice. Good luck!!!