This is much discussed in the Digital Music forum.
There are many different options. You'll need to do a little research to decide what is best for you.
You'll need a turntable/record player of some description, and an audio cable from the speaker jacks on the player to the sound card (line in jack) on the back of the computer.
Next you'll need some software to record the music as it is played on the player.
If you have Windows, I believe that it's possible with the sound recorder in Windows.
I use a program called Goldwave, which allows for editing of the files, once recorded to your harddrive.
A wave editing software is very helpful, since you'll probably want to remove scratches and other noise produced by the record player. It will also allow for changing the volume of the file, which you're likely to need. Check download.com for freebies.
There's a little learning curve to most of the editing software, but nothing extremely difficult. I'll leave it at that, since you're likely to get many more suggestions from others. That's just the basics, and a starting point.
p.s. check out the digital music forum.
Best of Luck