Hi Tyke,
The AMD chip in your computer is an AMD APU chip......................mea ning the CPU and GPU reside on the same die, just like the modern mainstream Intel CPU's are designed. This APU should be replaceable.
From seeing an all-in-one PC in Best Buy I don't believe there's enough room to add a discrete vid card in 1 of these units.......
I think the only recourse to keep using the existing box is to see if you can replace the existing AMD APU chip w\ a larger more powerful version to up the GPU power\capabilities........if the mobo\BIOS will allow. In addition w\ an APU upgrade a last resort would also be to increase the amount of onboard system mem to the max the mobo will handle as a lot of these units have single-sided DIMM modules installed (how they get the off numbered RAM capacities.....such as the 5Gb shown to be installed in this unit, most likely 1 3Gb DIMM and 1 2Gb DIMM, both single-sided). The APU may also be hindered by a lack of enough system mem to accommodate the increased graphics frame sizes from AHIII as well as all the rest of this computer's devices and so the HDD is being paged to death to make up for this slowing the entire unit down. You might also check to see if there are resources eating up CPU\GPU time which will take away from the necessary CPU\GPU resources needed to run the game decently and turn all off that aren't necessary to be running. If money is tight then what I've mentioned here will be your best way forward at this time but don't get your hopes up too high..........
Either\both of these options will help this box to perform better than it is currently but the risk of it not performing to your expectations running AHIII is still fairly high IMHO. You can only get so much from this type of computer performance-wise where gaming is concerned.
Otherwise what the others are saying will be the better, long run option to take...............get a new PC that is capable of playing AHIII (or other games as well) to your expectations and meet any other computing expectations you have......if gaming is 1 of the high priorities for you. You'll be happier in the long run.
Hope this helps you out.