Yes you can easily and safely run a car audio system off an ATX PSU, just not an additional secondary amp unless it's a very low powered one.
Use a PSU with an honest wattage rating so you know where you stand with available 12V current, and preferably one with a single 12V rail. Cut the connectors off the leads on at least enough leads that you end up with some margin, ideally (rated 12V current / 6A per lead) = # of 12V leads needed. Now do the same for the negative leads.
Twist and solder all negative leads together and the same for positive, and solder to the corresponding points on the radio wiring harness. If you don't have the harness you'll have to take the radio apart and solder leads to it, or source the harness which should be dirt cheap from a junk yard if it's an OEM factory equipment stereo.
On some PSU you might have an overvoltage shutdown situation if you don't put a bit of a load on the 5V rail. A power resistor is one easy option for that load, something like a 25W chassis mount 2 ohm resistor that you put heatsink grease on and bolt to an old heatsink, something like the following:
http://www.digikey.com/product-detail/en/KAL25FB2R00/KAL25FB2R00-ND/1646179 then just strip and solder one 5V and one ground lead to the resistor.
You might also want to wire up a SPST power switch to the green PS-On PSU lead and a black ground lead or if the PSU has its own power switch on the rear you can just solder those two wires together and of course you should cover all solder joints with heatshrink tubing to insulate them, and mount the heatsink with the resistor to something that can withstand the heat as it will be getting hot enough you can't comfortably hold it unless it is a fairly big heatsink... something roughly 1" H x 2"W x 2"D or larger should suffice, something like a heatsink from any regular power level desktop CPU made in the last dozen years.