All you need is an input jack which you connect to the back of the computer, exactly where you put the mic. Connect that to your record player. You can put it in the headphone socket or, depending on your stereo, where the speakers are attached. You'll then hear the music playing through your computer.
You then need something to record it, so download dBPowerAmp Converter.
http://www.dbpoweramp.com/dmc.htm
That will record the tracks as either MP3 or Wav files. The instructions are on the site. The MP3 converter is only on a 30 day trial though but Wav is better quality. I find that's the simplest way to do it for free. Once it's on your computer just use your normal way to burn the CD.
Hope that helps.