Well, it depends! The first important consideration: what style do you play / wish to record? For a classical performance, you need quite a different sound than for a folky dance tune, a jazzy improvisation or even a rock or metal lead role.
The main part of the differences in sound is the room component. For a classical recording, you want a good-sounding room/hall with a considerable amount of natural reverberation. That's probably not feasible for you right now, so I shan't discuss it further here.
For other styles, such a concert-hall approach generally isn't really needed or even desirable. Still, it's advisable to record in a room with pleasant acoustic character: unless you place the microphone(s) very close to the instruments, the room will be audible.
In my experience, it's impossible to get a properly "smooth" sound when recording string instruments with close microphones only. This probably has to do with the combination of the instrument's sound surfaces: only if all emission directions contribute to the overall sound (via reflections), it sums up to something with both good bow definition and pleasant body tone.
That said, even a close-mic sound has its applications; it tends to be pretty harsh, but the immediateness makes it possible to place in a loud rock environment audacious solo parts that would with more traditional sound just "drown" in the mix.
If you want precisely that, you need just one microphone. A large-diaphragm condenser mic (e.g. AKG C3000, low-budget Behringer B1) would be my preference here: it gives you a very full, powerful sound. But you might even try a dynamic microphone (e.g. the classic Shure SM57), this would give you an even harsher mid-oriented tone.
More likely, you should permit some natural room sound dissipation as part of your recording, with not-too-close mics. You won't get around condenser technology then, only this offers the treble-fidelity, linearity and noise ratio required for a really good "natural listening experience".
A single large-diaphragm mic, merely placed a bit further away, can yield quite a good base sound. However, the reverb part usually comes out much more efficient in stereo, so (keeping it simple) I would prefer a matched stereo pair of small-diaphragm mics. The Oktava Mk12 has become a quite popular still-affordable choice, and for even less money there's again a Behringer offer, the C2. There also are readily-built stereo microphones around; those are easy to set up (at the expense of flexibility, but that may not matter much to you).
Connecting to the PC
You need an audio interface with at least two built-in mic preamps incl. 48V phantom power (that's needed for condenser mics). There's a wealth of such interfaces available, e.g. M-Audio Fast Track pro, Focusrite Sapphire...
That's pretty much it then, just connect the microphones to the interface with XLR cables and the interface to the PC with the USB or FireWire cable.
Apart from the drivers for the interface (ASIO is de-facto standard on Windows) you need a DAW. For you application the free Audacity should be fine; should you need some more post-arranging, editing & FX stuff I'd recommend Reaper.