Going back to your original post,
since you are looking for the cheapest way possible, there's not much point in spending extra money on anything if you have the capability to get sound into your computer.
That is, yes, it's
feasible that you could record a demo with just your computer mic and reaper.
Will it sound very good? No.
Will reading up on, learning, and practicing the use of reaper (or whatever DAW you go with) help enormously? yes.
The thing is, before you go spending any money, record with what you've got.
Then, if (probably when) you don't like the sound of it, figure out what you don't like, search this site (and the many, many other sites like it) for information about fixing those problems, and then post questions about specific things you can't find answers to.
I know you're in a hurry, but, as was said before, the only way you're gonna get close to quality quickly is to pay someone who knows what they're doing to record you.
If you're the type who's deadset on doing it yourself (and I'm one of those people, so please don't think I'm trying to push you away from recording at home and towards paying someone else to do it), then, you know, do it.
DON'T DON'T DON'T spend money you can't afford until
a) you're sure you have to
and
b) you're sure it will improve your sound.
You are not gonna get the absolute best sound out of the equipment you have the first time you hit the record button.
It's like picking up a guitar for the first time, that happens to be a piece of crap, and thinking "man, I don't like how this sounds, I better go spend more money on a better guitar." The first time you picked up a guitar, it could have been a tricked out custom blah blah blah, it still would have sounded like crap, cause you didn't know how to play.
So, conclusion wise:
1) you are never gonna get professional quality out of the equipment you have.
2) buying new equipment is NOT gonna be the big step towards getting close to professional quality. Learning how it all works is.
3) I have diarrhea of the fingers in the mornings, apparently.
3 is important. 3 lets you know it all might be crap.
