There are quite a few roadblocks to overcome before you can turn this PC into a serious gaming beast.
The biggest ones:
1) PSU not strong enough to handle serious gamer cards
2) Case may have trouble fitting serious gamer PSUs and video cards
3) Processor not the best at handling add in graphics cards
None of them are particularly hard to fix, however, if you try to address just one thing you will likely find how interrelated they all are pretty quickly.
I would be surprised if you could manage to get a serious video card in that case, for starters. The CD drive is up and down because it doesn't fit sideways.
The lowest bargain basement case you might think about changing to that would give you room to hold what you need and good airflow would be a Cooler Master Elite 430 which will run about $50, but just for today there is $25 in rebates and discounts if you get it now.
That kind of case would be able to hold any video card and any PSU, so you wouldn't be stuck with tiny video cards and tiny PSUs.
A decent low end PSU would be an XFX Pro 550w, which will run about $65 with about $10 worth of rebates. That would let you run pretty much any single card you could get your hands on.
If you were looking to limit yourself to cards that have just 1x PCIE power cable connector there is the Corsair CX 430w that is $45 with $20 worth of rebates instead. It would be very aggressive to run this with a top end video card with multiple PCIE power connectors, though, even with the necessary converter cables in hand to allow it to work.
Then there is the card itself, a HD 6850 is about the lowest level that one can call a serious gamer card, and that will run about $150 with about $25 worth of rebates.
Then there is the processor. An A8-3870k would be able to squeeze a whole lot more performance out of that card than the A4 will be able to by a long way. That would run about $110.
To get Case (50), CX 430w (45), HD 6850 (150), and processor (110) would require an up front cost that's about $350. About 80 of that would come back in rebates in 3-6 months, though, for a net cost of about $270 in total, by my rough estimates.
Feel free to double check any/all of that.
Obviously, with the better PSU and a better video card, the cost would scale up from there. The XFX 550w and a 560TI in place of the CX 430w and the HD 6850 would add about $100.
Saying you weren't so much interested in going this far with it you could potentially just look at replacing the processor and the RAM. The same processor from above would be $110 and if you really want to make the most of Llano onboard graphics you really need more than 1333 Mhz RAM that comes standard on the Aspire desktop you have.
I am having a tough time finding the motherboard part number for your computer, but its an A75 chipset so it should be able to hold 1866 RAM I would think. That is generally what people should try to aim for as a minimum with a serious attempt to maximize the llano graphics. That would run about $50.
Those two things together would be about $160 and you would have quite a bit worse performance than if you spent the above $270, but it would come with quite a lot less effort and more than $100 less of a bite in the wallet too. Still, it would land you about in the middle of the two.