I've been doing this for the past few years, but only for LPs that are rare / out of print / not available via any other medium. As Mlsstl notes above, this is not that same as going from a master tape to digital as there are all the vinyl steps in this path. That said, some recordings, i.e., the Columbia 360 LPs, do sound incredible when captured this way. As Sevende suggests, take the output from your phono pre-amp directly into the input of your ADC. In my case I take the balanced outputs of my Xono pre-amp into an Apogee Rosetta 200 ADC / Symphony system. I use Peak software for capture but there are many good programs out there. And, as ART says, record entire sides at once and if you want separate tracks do that as a second step.
I find there is a different mindset to this than simply listening to LPs as you'll being creating (at some level) an "archival" copy. So, a few more suggestions:
1. Turntable setup becomes even more critical. As most capture programs provide some type of metering, by using a good analog test LP you should be able to confirm matched channel levels. More sofisticated metering, e.g., phase and spectrum displays will let you align things even more precisely. Of course, these must be confirmed with listening tests before you start recording!
2. Check the turntable speed right before you begin recording to ensure it's dead on.
3. Input level to the ADC is absolutely critical and will vary quite a bit from LP to LP. You'll want to set the input level as high as possible so that the very loudest passage on the LP does not hit clipping. Be careful not to confuse the loudest passage with a click or pop as these will peg the input meters.
4. Once levels are set and confirmed and you're ready to go give the LP one finally cleaning with your record cleaning machine. Also clean stylus before recording each side.
5. I record at 96KHz / 24 bit as you suggest. However, if you need to down convert a copy to burn to a CD or for iDevices, than I recommend you convert from 24 bit to 16 bit FIRST and then do the sample rate conversion to 44.1 KHz. This is counter intuitive but, to my amazement, many listening tests support this workflow.
6. Even the cleanest LPs will have a few pops and clicks when you capture them. NEVER use any of the automatic click removal / noise reduction programs to get rid of them! As good as some of these have become, they're still not good enough -- they all leave artifacts, e.g. robing the air from around the notes. If the pops and clicks bother you then go in and paint each one out of the waveforms by hand.
7. If you want separate tracks create them after you've done the above.
8. Be prepared to spend a lot of time doing this. Having spent over a year getting every aligned correctly and sounding it's best, I find it now takes me 4 to 6 hours per LP.