That 6150SE is integrated to the motherboard, right? According what I found the computer has a 300W power supply. Apparently it also has a pci-e slot, but checking has never hurt anyone.
A quick calculation with the
Extreme Power Supply Calculator showed that a Radeon HD7750 would run with your wattage, although the AMD site says you'd need 400W. Compared with titanic's suggestions the requirements are on the same level both in the calculator and AMD's specs. The 7000 series needs less juice than its predecessors. Older high end cards might perform well, but they are energy hogs.
If you look at this
PassMark Low End GPU chart you'd find your video card at the 21 points level, overhauled by low end cards of last decade. 5570 got 699, 6570 got 747 and 7750 got 1584 points in the chart. The top cards scored over 5500... Of course all such charts can only be considered as suggestive, but definitely almost
any card looks better than the one you currently have.
A more powerful card may shorten the life of your PSU, but maybe you could think of it as a part payment new build?