I also added a short sound clip, so you can get an idea of the sound quality. I didn't run it through any filters, so that's what it sounds like directly off of the record.
Step 1: Stuff You'll Need
2.) A Phonograph (a.k.a. a Record Player, I'm using a Beogram 8002)
3.) A Phono pre-amp (In my case a Rotel RA-8408X)*
4.) A Stereo RCA Audio Cable
5.) An RCA to Mini-Jack adapter
6.) A Computer with recording software
You can easily pick up records at places like DI and second hand stores, or the like.
If you don't have a Phonograph, you can find them online, in second hand stores, ect.
The RCA cables can be easily acquired at Radio Shack, other electronics stores, or online stores (Monoprice http://www.monoprice.com/home/index.asp is my favorite)
Pre-amps can be found online, the cheapest one I've found is this one http://www.mcmelectronics.com/product/40-630. The pre-amp is necessary because of a special equalization curve, called the RIAA EQ curve, that was applied when the record was recorded (RIAA stands for the Record Industry Association of America). The curve limits the lower frequencies and boosts the higher ones. A pre-amp then limits the high frequencies and boosts the low ones, creating a good-sounding replication. If you didn't use a pre-amp, you'd end up with a poor quality recording.
Finally, on to the the computer and software. Basically any computer will do, you just need to have a line-in input, and sufficient hard drive space to store your recorded music. Having a more beefy computer helps with ease of editing and exporting the final product. The software I use is called Audacity http://audacity.sourceforge.net/, it's free, cross-platform, and open source. Plus, it comes with some handy plug-ins that you can use to clean up your final recording. I'm using the beta, if you're a beta kind of person then go ahead and use that, but for others I would recommend the stable release.
*Recent releases of Audacity include the RIAA EQ Curve (and many other EQ curves) with the Equalization effect by default, so a pre-amp is not necessary.