Hi there and welcome aboard.
Let me answer your question quickly. As long as you have a PC that's equipped with a soundcard, it's perfectly possible to record analog sounds on it. As you indeed don't have any RCA bus on your soundcard, you need to buy a convertor cable from 3.5mm stereo jack to 2x RCA. Radioshack has these cables.
You can't just hook up your turntable to a soundcard. To do so, you need a preamp. Terratec sells these preamps for about $70.
There's a cheaper way though... I almost presume you have an amplifier that supports phono (turntables) and has either a lineout or a headphones out. What you do is connect one of these two outputs to the line in of your soundcard and record the sound that is coming from the amp. Now, the amp works as a preamp for the turntable and the sound can be recorded.
I hope this helps a little. If you got any more questions, feel free to ask! And euh.. good luck!