This is a brief example what one can do with a Raspberry Pi computer, a PC running Linux. It is by far not the only example of converting an old radio to an Internet radio. Please search You Tube for more examples.
Also search the web for better explanations of a Raspberry Pi.
Logitech Music Server info here: http://wiki.slimdevices.com/index.php/Logitech_Media_Server
PiCorePlayer2 software here (the actual player): https://sites.google.com/site/picoreplayer/home
A quick easy way to install Logitech Media Server on your Pi: http://www.squeezeplug.eu/