Video and sound recording are two separate things. Consumer camcorders do both in a single machine, however when it comes to more serious music recordings often it is the sound portion that is lacking.
Getting a better video camera may result in better video, but not necessarily the sound you desire for critical listening to music. You may need a separate sound recording setup. You definitely need something better than a computer mic to capture sound for music.
There are flash recorders that do sound with internal mics (such as the Zoom H4n), these should be good enough to make a demo recording. News reporters use these flash recorders for broadcast, as well as people that shoot video with DSLR's who want an independent sound recording.
If you want better quality than what these offer you are probably looking at investing in a more dedicated recording rig with individual components. Mic, preamp, and recorder is the basic idea.
For now, see if your equipment has some sort of "auto gain" feature. If it does, turn it off. It's fine for speech, but it will compress the dynamic range in music and cause any manner of artifacts. When doing music recording you want to have manual control of the gain.
Topic: Video Recording