Very impressive cover! As mentioned before, you could really use a better microphone. Depending on your budget, there are a few different microphones I would suggest.
Shure SM57 - Great for vocals, but can be used for instruments as well.
MXL 990/991 - These come in packages from places like Guitar Center, Musician's Friend, Sweetwater, etc. They'll run you about $100 or so.
If you're looking for higher end microphones, look into Blue mics. A highly praised mic is the Blue Bluebird, I've had the pleasure of recording my guitar with it and I've heard numerous vocalists use it in a professional setting, it has never been a disappointment. Unfortunately, that'll run you about $300, not including the interface you'll need in order to actually use the thing.
All total you could be looking at anywhere from $200 to $500, depending on how much you're willing to spend in order to get a top notch sound.
But remember, it's not always the hardware you're using. Microphones aren't magic. It's going to take a keen ear to find that sweet spot in the mix.
Best of luck to you. (:
5 and 5. Favorited.