Agree with Guitarplyr82.
If you are doing this for anything less than network television distribution, get a program that does video editing for your computer. I used Sony Vegas for my vid and it worked fine for what we were trying to accomplish. It allows you to have multiple video and audio tracks. You can use transitions to cut up the singing into shorter clips with different camera angles and that will give an illusion that you are not lip-syncing. I think that if you are unfamiliar with the song that you are singing, the worst thing you could do is try to sing the entire song in one take in front of the camera. Get familiar with the song and do a few takes.
As Guitarplyr82 said, once you fly that raw footage into your computer, you can take out the audio from the recording that you just filmed. Just about any video software I have ever played with will separate the video from the audio into separate tracks. Import your song into the program and delete the audio from the video shoot making sure that the two audio tracks are sync-ed before you delete the audio from the video shoot. Lots of work, but also lots of fun. Good Luck!
-C