If your fx8320 is only running at 4ghz, I would suggest OCing it a bit further first. You should really be able to get 4.4-4.6 fairly easily with most coolers.
If you upgrade, there will certainly be an improvement, but as has been said, it may not be worth the expense. Only you can decide that unfortunately. Try looking at some other the benchmark threads, they may give you some idea on what to expect.
If you are looking at a 6600k then a cheaper option is the 6400 which can be overclocked to the same sort of levels with the correct mb (8pack bundles are a great option). With either of these cpus, you will be limited to 4c/4t (no ht) but they are strong especially for gaming.
5820k will offer you similar performance but has the added benefit of more cores and HT (6c/12t), which although it will not make much difference for most games currently available, it will help with other tasks you may be using the system for, and its probably a bit more future proof.
