hd 5770 will run on that psu fine(i saw it as low as $80 AR),there's sometime hd 6850 pop up for $120 AR and it's pretty nice too but you have to camp for that ,but i was shocked when i saw the title said $1010
you should close the other duplicate thread. and correct the gfx card budget.
that psu might not be able to deliver 600w power. it has no power factor correction, it will draw more power from the wall socket than other efficient psus. http://www.hardwaresecrets.com/article/Cooler-Master-eX...
the 5770/6770 might be able to run on the psu. if it's old, you should change replace it with a better, 80+ rated, active pfc psu like corsair builder cx 500 v2 or seasonic s12ii 520w or antec neo eco 520c etc. low end psu components age faster and can damage reliability of the system.
Yeah it would run well, It would deliver enough power for just a 5770. I don't know in what country are you in, so I don't know what's the good price on your country, but in US they only have one better with $110 budget, which is 6770 (for new cards) So I guess you're OK with that.