The i5-2320 is still a pretty good CPU. It suppose it's only real limitation is that you cannot really overclock it. The i7-3770k is not really worth spending money on if you are only playing games. The only game that benefits from Hyper Threading (HT) is Battlefield 4 and only in multiplayer mode. Additionally, if I remember correctly BF4 only really benefits from HT in Win 8; the performance gain with HT in Win 7 is rather small. That implies BF 4 has been tweaked to run better on Win 8 than Win 7.
Assuming you have a Z77 motherboard, you will be able to overclock the i5-3570k which better suits your needs. However, the 1st thing you should do is buy a new power supply. I generally prefer Seasonic PSUs since they are basically in a league of their own. But their PSUs are usually more expensive than other PSUs especially since they do not go on sale as much as other brands.
If you do not have a Z77 motherboard (or any Z7x series), then there is no point in buying a "k" model Intel CPU.