Your 600w PSU should be fine, since the 5770 only has a max TDP of 108w. So running two in concert (unless you overvolt) will run no more than 216w under a full load.
Unfortunately, you'll need a PSU with 2 6-pin PCIEx connectors to power both cards (each card has a 6-pin PCIEx connector).http://ncix.com/products/?sku=42532&vpn=CMPSU%2D650HX&manufacture=Corsair
It's a bit more expensive, but I've heard good things about Corsair Powersupplies.
It's also modular, which is a big plus. If it's too pricey, let me know and I'll help you find something
a bit more cost effecient.
If you were going with something bigger, I would recommend a larger powrsupply, however,