Sorry to say, you'll need a new PSU before considering a modern graphics card.
With adding a current graphics card to a pre-built machine, one issue that invariably comes up is the sub-standard PSU's used. Your machine appears to have a 250 watt PSU,and I wouldn't trust it to be able to consistently supply even that wattage.
currently your system draws (under full load, approximately)
CPU - 65w
mobo/RAM - 50w
Hard drive - 30w
optical driver - 15w
CPU/case fans (5w ea) - 15w (covers 1 cpu fan, 2 case fans)
wit the rest of your system specs, a 7750 or 7770 would be a solid pairing with your CPU and target resolution. The 7750 can draw up to 75w on it's own (the 7770 pulls more, and would need a 6 pin cable which you PSU almost certainly doesn't have). drawing 250w from that PSU is dangerous to put it mildly.
I reccommend the corsair builder 430whttp://www.newegg.com/Product/Product.aspx?Item=N82E16817139026&Tpk=corsair%20builder%20430