Unfortunately HP doesn't seem to be inclined to tell us what power supply they stuffed into that thing, so it's going to be hard to make a recommendation without you checking to see its wattage, and preferably how many amps are on the +12V rail. If it is a 300 Watt unit, the best you can get is the Radeon HD 6670 or the slightly less powerful 5670. Anything more powerful would probably be bottlenecked by your CPU, and would require a better power supply than the one HP probably gave you. If it is a 250 Watt PSU, the best you can safely run would be the Radeon HD 5570. Any of these cards would be significantly better than the integrated graphics currently on your system.
Even with a better power supply, the best you could really do would be the 6670, your CPU will hold back just about anything more powerful than that. Maybe the 5770/6770 would be a viable option, but you probably won't get full performance out of that card. Since you have a prebuilt, overclocking isn't an option. Unless you are willing to do a CPU upgrade, and your board can actually support a faster CPU, there's no point in aiming any higher than the entry level gaming cards.