I wouldn't place any faith in that power supply. If you're going to do an upgrade, I highly suggest getting a higher quality PSU while you're at it.
RAM is pretty expensive right now, with a $200 budget, you'd lose almost half of it to RAM, motherboards aren't cheap either, and the G4560 is around $75 right now.
Could you scrounge up more money by selling your current rig except the gpu and hard drive? Even an extra $50 into that budget will likely go a long way. Bumping up the budget to $300 will make this way easier.
On a really tight budget like yours, I'd probably get a used i7 920 + X58 motherboard combo on ebay for about $170-200, reuse your old RAM and HDD, and get a Seasonic M12II PSU. And then get a CPU cooler with whatever budget is left and overclock the i7 920.
Or better yet, an old Nehalem Xeon like the W3540 (essentially an i7 940) and a used X58 motherboard and a new PSU. Those CPU's are like $5 on ebay right now. They also overclock pretty well. I got mine to 4ghz~ish from its 2.93 base clock on bus speed alone. Pretty beastly, but I don't know if that's truly an upgrade, or whether it's more of a side-grade.
TL;DR Best solution is to save up more money or come up with more money. Money talks, eh?