Well first off, do you have a turntable that you can hook up to RCA inputs without the ground or a preamp? Not going to get far without that. You also need a Y-adapter or similar to connect the red/white RCAs to the headphone-sized input jack in the back of your computer.
Basically, any sound recording software can do a decent job. Look for ones specifically for recording analog that can take noise (hiss), crackles and pops out. You also want something that'll let you edit tracks (so you can record continuously) before finalizing.
Of course, the software can only make your recordings sound as good as the records you use, so make sure you wipe and clean each one (circular motions, not like a CD) before recording. Remember, you're live when you hit that button.
Personally, I used a program that came bundled with Windows Media Edition. Does the trick.