What power supply do I need to power 4 peltier plates with 5A 12V each?
Short answer is you need around a 25A 12V  (300W) supply. A PC supply would work for that.

However, the long answer is those things are really finicky and die easily. You really need to use current and temperature sensing for each plate, and getting rid of 60W of heat per plate is no simple task. Making sure the heat exchanger is intimately touching the surfaces with a good heat exchange medium between them is also key. 

Also, they need to be sealed, any moisture in there will also kill you.