The main part would be getting the L/R RCA outputs to the stereo mini jack input on your pc. (Radio shack, walmart, any electronics store has these for under $5 for the adapter and under $10 for a 6-8ft RCA cable if you don't already have one) Then any audio recording software can record it.

Open the software, have it monitor the LINEIN port. Start recording and then play a track on the LP. When the track is done, stop recording and then edit out the dead space at the begining and end of the track.

I've used Cool Edit for a long time to edit sound files, it's way overkill for what you need, but Sound Recorder (built into windows) might not be enough.