The Linn would work. You also need a phone stage either separate or in a preamp.
You can use a sound card in a PC, but the quality is often not very good. You can get an external A to D converter from anywhere from $100 to many thousands. You can use a handheld like the Tascam DR-5 ($100), or the small Tascam DR-60D ($150). Either will do up to 24/96. Oher common ones are the HRT LIne Streamer ($350), the new Korg DS-DAC-105 ($600), Tascam DA-3000($1,000), Benchmark ADC1 ($1700) and so on. If you purchase one, you should probably be sure it is returnable just in case it does not live up to your expectations. Pro music stores - even Guitar Center - have A to Ds available. You should probably plan to record at 24/96 or possibly 24/192. Many people find it sounds better than 16/44 but that depends on your system and your ears.
There are 2 different types of A to D converters available. Some store the digitized data on a disk drive and you then transfer the data to the PC either by moving a flash card or over usb. Others connect directly to the PC via usb and the digitized data is sent in real time to the PC. The later ties up you PC when you are recording and requires you have it near you turntable. Just something to think about.
Audacity is used by many, as is Vinyl Studio ($29). Vinyl Studio has the advantage of being tailored for vinyl recording so the workflow is better. It also looks up album and track data on line, which saves a lot of time typing.
When you are starting off, you should plan on 30 minutes or more for doing track breaks, tagging and cleanup. You can do it faster once you are experienced, although some will take a lot longer if they have a lot of damage.
If you only have a few to do, you might also look for a service to digitize them.
