I'm not quite as negative as others. I don't really see your CPU as much of a problem. 820M otoh is not so hot. Try it. But a gaming PC it isn't.
Best way to get a cheap but decent gaming PC, is to build a desktop. Some people get carried away and buy all the ridiculously hot stuff they rave about in hardware forums. That's why the general perception is that a gaming desktop is expensive. It isn't. You just have to be modest when you pick components. The most important thing is the video card. And regardless of numbering system and era, it's the second digit in the graphic chip's number that is the first clue to the performance it's intended for. So you see, your 820M - second digit = 2 - was never meant to blaze things anyway.
Nvidia 960 and AMD 280 seems to be good buys currently. As long as you get parts that fit together (correct sockets, size factors), and your PSU can power your graphics card, all the other parts can be quite cheap. Cheap case, cheap standard ram, straightforward non-sli/non-crossfire motherboard, without integrated graphics, cheap common harddrive (coming from a laptop, you obviously can do without an SSD), simple optical drive (you don't have to write discs). Do get at least a four-core CPU, but otherwise the CPU is almost indifferent. Don't even think about over-clocking. There's only one thing where *quality* is an issue and where you should avoid the crap, and that is the PSU.
A warning about cases! Do see to it that the pressure inside the case doesn't drop much at all (due to the various exhaust fans). If it does - it doesn't take much of a pressure drop - then the PSU will not get enough cooling. Big inlet openings is the key. You can even take off one of the covers in front of an empty large drive bay (that's what I usually do).