If you have to buy a turn table, then yes, might as well buy one of the USB turn tables designed to do exactly what you want to do.
I converted a ton of my old LP's using a turn table with a built in pre-amp, most turn tables do NOT have a pre-amp, that is why the amplifier you used had a special input for the turn table, it had the pre-amp built into the amplifier. You can search around for a pre-amp that you can plug in between you're turn table and sound card.
You need to get an adapter cable, to convert the different plugs, radio shack is a good source. Then you connect your Turn-Table to the Pre-Amp (unless it has a built in Pre-Amp) then to the adapter cable then to the "Aux" input to your sound card. You can use Windows Sound Recorder to record your albums, but spending a few bucks on a software program designed for this will be worth it. I used "Cakewalk Pryo", but that was years ago, which is like decades in the world of software. But it had noise filtering, tools to help split the album up into tracks, convert to MP3 or CD format, etc, etc...
If your starting from scratch, then the USB turn table is probably the way to go, it may include software tools to help as well.