Well, The simplest, dirtiest, easiest route is to get a 1/4"(big) --> 1/8"(small) adaptor and plug your bass directly into your computer via the computers external mic input or audio line in.
then you have two options(that I know of)
Amplitude live! or
garageband(mac only)
both these programs provide you with software amplification, quite well I might add. The latter also provides you with 64 tracks to record with and a wealth of virtual instruments.
Amplitude live is ~$50 if I remember correctly, Garageband is $50(part of the iLife suite, which includes 4 other apps) it's also free with a new mac.
So that's one route.
the other route, which is a little easier in some respects, but it will also cost a little more.
bass-->preamp-->1/4"-1/8" adaptor-->computer
this will give you the amplification before you get to the computer, from there you can use protools free, or cool edit pro, or any other free sound recorder and you'll get an amplified signal that should be more or less pretty clean(assuming you have good cables/adaptors)
Additionally, if you have a floorboard that can provide an amplified signal(can you wear headphones with it without your amp in the mix?) then you can use that.
basically anything that will amplify your signal before it hits the computer, so that your computer gets a strong clear signal, and you're set then you can use free sound recording software.
Price would vary, but from my knowledge, it'll set you back at least $70 for some form of amplification, that's optimistically too.
your amp head might have some form of line out, if that's the case, you should be able to use that in conjunction with and 1/4"-1/8" adaptor and then you'd be set.
you could also get a beefy sound-card, the audigy pro or something, those are like $200 I think, but they have 1/4" built in, and can work as a preamp.
Additionally, you can get a USB or firewire DI box,
www.m-audio.com makes some good ones
www.edirol.com does too.
