The HD6770 is a little stronger, BUT...that "ATX 450w" PSU looks suspiciously like junk, perhaps good for no more than 300W. If that is the case, it will run a HD7750 but possibly not a HD6770. Particularly if it does not have a PCIe power connector, it most likely lacks sufficient amperage on +12V for anything stronger than the HD7750. Do not use adapters with a PSU-shaped object; the story will likely end with smoke and sparks.
There is enough voltage/amps to the 12v rail for a 7770 even. I think what Onus is saying without a established power supply most of the time the quality of power supplies tends to be quiet poor whether it came with a custom case or from a retail computer they are normally on the very low end on the totem poll.
Well, even if it's overrated by 20%, it still ought to handle a HD7770. Of course, overrating is not the only problem common to cheap PSU-shaped objects. It may have sorely deficient filtering (input and/or output), significantly affecting how "clean" the output is. Dirty power can damage attached components. It's your call of course, but getting a quality (high [claimed] wattage <> quality) PSU would be a good idea. On a budget, the Antec VP-450 or 380W Earthwatts are good, reliable choices that can easily handle a HD7770.
What about 6670 or 6750? Cause i am in tight budget here. So it's quite impossible for me to upgrade my PSU. I can only afford a GPU. Will these card handle games like FarCry 3,AC 3, Hitman Absolution at 1336 x 768 in high settings?
At this point looking at your psu you have 2 choice hd 7750 or hd 6670 ddr5.both do not need 6pin connector.you just have to plug it in pci exp slot.i would wait and save up more and get hd 7750 or if you can't get then hd 6670 ddr5 is way to go.