I'd say almost no one would make a video while playing/recording live. It happens -- for example, during a live concert -- but most likely they'd have multiple cameras to record the whole thing from at least four or five angles.
And for the audio, they'd take a multi-channel feed that's at least as complex as what the live engineer is working with, then remix it afterwards until they have it sounding the way they want.
For someone doing a home made video like you are talking about, jeena, you might want to do a live thing where you mic everything up like you would at a gig, then take a stereo feed from your soundboard and send it to the camera and rehearse a bit until you have the levels right. Then you could to a 'single shot' video of you and your band playing the tune live, recording both video and audio at the same time. Another way would be to feed a recorded version of the song both to the camera in stereo and into some monitors for you and the band to listen to, then 'air-band' to it or come up with a script you could act to while the tune plays. Again, this would be a single uninterrupted shot video.
If you want a multi-shot video, you could shoot without recording the audio (just use it for reference while shooting) and then edit pieces of the various takes together and match it up to the original sound track making sure everything is in sync. A lot more work, of course, and it requires editing capabilities.