I just got done doing a ton of this for my dad. He had a bunch of old cassette tapes that he wanted on CD, not records, but the same principle.
I recorded them onto the computer hard drive through the soundcard (Mine is stereo, but some no-name thing), I then cleaned the hiss and background noise out of them with Nero Wave Editor, one of the apps that came with my CD-RW drive. After all of this I burned them to CD. It turned out well, in some instances the CD's sound better than the tapes because there is no background noise and hiss.
Good Luck,
Mike