One of the biggest issues I have seen, or rather heard, before I knew any better, was that record players need to be taken from a 'phono' level to line level.
If there is not a phono pre-amp built into your record player you need to run the audio through something to bring it up to line level. A phono preamp can be had at Radio Shack for about 25 bucks it looks like:
http://www.radioshack.com/product.asp?catalog_name=CTLG&product_id=42-2825
My A/V receiver has a phono input on it, as well as line level outputs that I can take into my sound card. Your sound card will have line level inputs on it as well, or at least it should. I have seen very few computers without line level inputs.
You need a RCA to 1/8" mini-stereo adapter plug to go from the phono pre-amp to your computer. This is overpriced (read: Monster) but you can find similar at your local Radio Shack for under 10 bucks.
http://www.radioshack.com/product.asp?catalog_name=CTLG&product_id=15-2473
Now, your record player, to the phono preamp, to the line-in input on your PC. There are programs readily available that can help with the recording process as well which can simplify the process and help keep it clean. A quick Google search will give you a TON of software - the first hit I got was this:
http://www.blazeaudio.com/products/tapetocd_softpack.html
The one time I tried it, I didn't use a phono pre-amp and now my copies sound like crud. Nothing at all like the originals. I am going to start over and do it again, only this time I will have a phono pre-amp in the setup and my results should be much better.