Possible Upgrading

$300 probably won't cut it. Very likely, it's a socket AM2 board, so you'll need to replace it if you want a stronger CPU.
Better to find out what is salvageable, and go from there. You'll probably have a case, fans, and an optical drive. The PSU will likely be ancient, and too small to be useful, particularly after factoring in capacitor aging. You may have a [temporarily] usable hard drive, but it may be IDE or SATA 1.5Gb/s.
I'd suggest looking at this cycle's $500 SBM build as a starting point. Subtract what you've got, then add what you may not, like a copy of Windows. That machine was remarkably powerful for having a HD7850 in it; you could still play on "decent" settings if you dropped back to a HD7770 if necessary.

Well, you're a long way from the top, but there are a few things you might be able to do.

First, the processor. While its not that bad (I have a 4600+ myself, and it handles games quite well), its old, and bottlenecks easily. If you get lucky, you might end up with an AM2+ motherboard. Different from AM2, the AM2+ motherboards are able to take AM3 chips, so you would be able to put an AMD Phenom II 955 into there. If you are not so lucky, you might be stuck. You might be able to overclock that processor though.

As for the GPU, it depends on the processor and the power supply. You can put a GTX690 in there if you so please, but its going to perform very slowly, being that the processor is too slow. We don't know what the power supply is, but once you find out, we can decide on that.

Other things you might want to upgrade are the RAM. 4GB should be fine.

What resolution are you playing at? Firstly, I would recommend trying to play on it as is.

These are just some hypotheticals, but say I can't upgrade the processor, what card should I look at for just a gpu upgrade?

Say it's an AM2+, I was looking at a Phenom II 965 BE, would this slow down a more high powered card such as an HD 7850?

I can follow most things up until PSU. I get lost in that territory. I would assume this guy most likely has the minimum safe wattage he can. In 2 days I'll actually have the thing, then I'll know for sure.

The 955 shouldn't slow anything down short of a 7950.

If its not an AM2+, you really cant upgrade the card too much. Maybe up to a Radeon HD7750 or a 7770, but then you will start to see bottlenecks.

if your board supports am2+ then get a phenom II x4 most am2 boards do am2+ but doubel check if you can. mine is an am2 but my bios wont do am2+ (actually any other processor than the one it shipped with)