You need some recording software on an PC with a sound card in it. The freeware Audacity works very well.
Typically turntables connect to a pre-amp and then from the pre-amp to the amplifier via phono outputs. Use a twin phono to stereo audio jack cable and take the output from the pre-amp to the line-in jack on your PC's sound card. That it, key up the LP, press record on the Auadacity software and let the LP play.
If your turntable is built into a hifi system unit, then just use the headphone jack on the hifi and take that to the line-in-jack on your PC's sound card. You may have to play with the volume levels in the software if you use this route.