There's plenty of free audio editing and recording software out there. Take a look at
Audacity - it's pretty good. You may be able to clean up your recordings a little bit with this, but on the whole it's a much better idea to get the best microphone you can afford. There are plenty out there for under £100 - have a look at this thread for more info:
microphone advice.
A friend on the forum uses a brilliant little mic - I forget the name, but I'll either get him to post it up here later, or when he next comes over for a jam, I'll jot the name down and post it up. Basically, it's a multi-directional mic/ recorder. It records to high quality MP3 straight into its own memory. You then plug it into your computer and download the music just like getting data from a memory stick.
As a rule though, the better the quality of sound in, the easier it is to work with. Even a simple stage mic would be better than a webcam mic I would have thought.
Let us know how you get on