I'm pretty sure they just record using the microphone and the video at the same time, and then later edit in the audio from the microphone into the video using some software like Vegas or Final Cut. Or the cheap poor person version of those programs, which I'm not sure what those would be... I've never done any vocal things like that, but a few of my friends have, and that's how they did it.
So basically, just cut the audio from your camera and replace it with the audio you recorded into your DAW using a video editing program.