You can certainly "thicken" the tone, either with a real voice mult (sing it again, trying to sing exactly the same way - which won't happen, yielding a close doubling) or by adding a tiny amount of delay to the copy - along with any other adjustments (EQ, panning, etc.) or effects you want to try adding to it. It will give you a stronger lead vocal in spots that need it, but in general, try not to over-use it as it can get annoying to listen to for a whole tune.
Being in that exact same situation (alone, with a pretty lousy singing voice and can't sing harmony that doesn't scare children and old people) I tried pitching stuff up to create harmonies once with Audacity and the result was a backup group composed of The Terminator (lower parts) and The Chipmunks (higher parts). I finally let modern technology do the job, originally with a TC Helicon "Harmony G" stompbox and I recently upgraded to their "Voice Live" stomp box. These gizmos are pretty amazing. They aren't cheap, but made a tremendous difference in my recordings and they're pretty easy to get the hang of. If you're stuck recording vocals by yourself, they're really neat tools to have.
Harmony G samples:
http://webpages.charter.net/tbradshaw/M ... amples.mp3
First effort with the new Voice Live:
http://webpages.charter.net/tbradshaw/M ... See-er.mp3
) I need to try and come up with some way to punch up vocals in my recordings.
