For a more simple video, record a few test runs and adjust cameras, mics, amps, speakers, whatever your using to find what creates the best sound. For me personally, I play electric and I like to play it loud. I had some speakers playing the backing track near me, perpendicular from the camera and loud enough that I could hear it, but low enough that it wasn't picked up by the mic too much (which was just the camera's built in mic). My amp was just a few feet away from the camera, but facing me. This way, it was picked up by the mic well, but it wasn't blowing it out. Later, when I edited the video, I put the back track on top of the video and set the volume so that everything sounded nice.
Now it sounds like you don't have all that equipment. If all your trying to do is play some acoustic and sing along, you should be able to do that just fine with just your camera and video editing software. I wouldn't even bring audacity into the picture.
However, If your going professional, nice backdrop, good lighting, all that, Then yes, you might want to play with audacity a bit. Since I don't know exactly what you have in mind, the simplest example would be the acoustic sing along, if it could all be done in one take. In this case, set everything up, and record the audio and video at the same time, render the audio from audacity (after editing it to your satisfaction) and put it over the video after deleting the video's original audio. You'll just have to do your best to get it in sync.
If it is more complicated than just playing and singing (harmonies or whatnot) what I would recommend is you record the whole song with audacity start to finish until your satisfied. Then record yourself playing along with it. Record playing all and with every part and harmony, everything you want in the video. Put them in your video editor, delete all the audio, slap your recording in there and tweak everything until it's in sync.
Wow, sorry that was so long. But I hope it helps