HD 5770 is a good choice. But I really doubt that 300w PSU of yours can take it. It needs at least a good brand of 450w PSU with 1x 75w 6-pin connector. etc; seasonic, antec, corsair.
I used this to calculate the PSU:http://extreme.outervision.com/PSUEngine
It says it needs 293 (recommended) or 243 (minimum).