Is there a way to build a software defined radio using the PC's DSP chip?
What PC DSP chip are you referring to?  I don't know of any DSP chip inside of commercial PCs that's capable of performing software defined radio.  It is possible to do the necessary processing for a baseband bandwidth of a handful of MHz with the CPU, but there is no getting around the need for an analog front end.  A device the the USRP has a decent FPGA with enough DSP power to handle of up to around 50 MHz of baseband bandwidth along with the ADCs, DACs, oscillators, up and downconverters, and amplifiers necessary to actually work like a radio.  You will need something to provide this functionality.  

If you just want to receive, I would suggest looking at the rtlsdr project.  This project uses a TV receiver dongle that has about 1 MHz of receive bandwidth and just outputs the raw baseband samples.  It's probably the cheapest possible SDR receiver, but it has a very limited baseband bandwidth.