To OP : I did it a long while ago.
HW wise, remember that your turntable outputs very low levels, and frequency response is skewed (on purpose) by the RIAA correction. You will need a phono preamp to raise the level and apply the RIAA correction. The cheapest phono preamp is any old stereo with phono input and tape output.
So, connect your turntable to the phono input of the stereo receiver/integrated/preamp (the stereo).
Connect the tape output of the stereo to the line input of your computer (often the blue one). You will probably need a stereo RCA to stereo 1/8" jack cable. Those cables are easy to find in a Radio Shack.
Optionnally, a second of those cables can be used from computer's green line out / Front Left/Right output to the stereo's tape input, allowing you to listen your computer on the big stereo.
Using appropriate software (Audacity is good and free), record the whole face of the disk as a single long wav file, then cut and store individual songs, naming and tagging them appropriately.
Do yourself a favor, even if you always end up listening to those records as mp3 on your iPod or Zune device, do keep a lossless encoding copy of your rip as the high quality source, be it as .wav (does not support tags), .flac (universal, available on all platforms), .ape (MC native), apple lossless etc... as it WILL come in handy one day and you won't want to rerip everything.
It will take time to do it correctly, but you will have to do it just once...
PS: the links submitted by Vincent are quite interesting !
Cheers !