The quality of the turntable always makes an impact on the quality of the final result as do the other electronics you'll need as these are your source.
Turntables don't deliver a high enough output level for you to be able to record directly from them without some form of pre-amplification and the quality of the preamp will also impact on the final sound.
You can record the output from your HiFi system or there's a method where you can use a cassette recorder as a workaround but life starts to get complicated and potentially expensive if you don't already have the equipment or the cables.
It took me a while finding this but I knew I'd seen a reasonably cheap USB turntable a while ago that got good reviews.
Have a look at this review from Computer Shopper magazine.
USB turntables have all the necessary pre-amplification and RIAA audio equalisations built in and they're a relatively cheap option that provides good results.
As has been said above though it's probably not worth the effort for stuff that has already been re-released on CD.