http://computer.howstuffworks.com/power-supply.htm tells you the computer power supply will indeed work.
http://pinouts.ru/Power/atxpower_pinout.shtml says that Black on the radio goes to Black (pins 3, 5, 7, 13, 15, and 17), any one will work. And radio Red attaches to computer power supply pin 10 yellow.
Now the radio's clock should come on. DO NOT TURN THE RADIO ON WITHOUT ATTACHING AT LEAST TWO SPEAKERS. Some radios will burn out if they are turned on without speakers attached.

Now, which speakers? Use car radio speakers. Not some leftovers from a boom box. Some inexpensive speakers are almost as bad as no speakers.

Go to a junk yard or car radio store and get a car antenna or at least a plug for the back of the radio. http://ths.gardenweb.com/forums/load/electronics/msg0521100331318.html here is more information than you wanted to know about the length of wire for AM and FM radio antennas.

Have fun!

Some more usefull discussion here:

http://www.forumapex.com/other_hardware/96593-turning_a_car_stereo_into_an_in...