After some research, I've found that this achievement is apparently kind of glitched. It seems to pop at random times, whether the person is singing or not. I've read reports of people trying to legitimately trying to sing the lyrics to these songs and still the achievement won't pop. I've read other accounts of someone talking to a friend while playing and having the achievement pop in the middle of the song.
How I got the achievement, however, was to start a playlist on my laptop while dancing and put the laptop close to the kinect for the off chance that it would think I was singing. It apparently worked because the achievement popped after about 30 minutes of doing this (at the end of a song). You may want to try this, unless you want to attempt singing the lyrics legit, (hope you're comfortable with your sexuality, men).
Edit: After re-reading this solution several months later, I thought maybe it has something to do with hitting a certain number of the vocal queues. The ones where you're supposed to shout something while doing a pose. Just a thought.