If you are on a budget, AMD is the way to go. An AMD fx 8350 will only cost around $150-170 and it is an 8 core, 4ghz cpu
If you can't spend that much an AMD fx 6300 is quite good for it's price, which is around $100 and it is a 6 core, 3.5ghz cpu.
Both are overclockable.
I would go with the 8350 if you can afford it but both are viable options. Overclocking would also make either of them a better choice and would help you keep up with the latest games better than stock speeds, but that depends on your PSU. If it is under 450W I wouldn't recommend overclocking.
If you want to go with an intel I would go with an i5 4460 3.2ghz, and it is $180. It is not overclockable so it might be better to go with AMD as it is better for your price range, as the AMD products performance is better than intel up to the $200 mark.
Just remember whatever cpu you choose the motherboard has to be compatible with the cpu, so if you are going AMD you would have to get a motherboard which is compatible with AMD, AM3 is usually the type of MOBO which is compatible with AMD whereas with intel it's usually something like Z97 or for the i5 4460's case, it has to be a motherboard compatible with the intel cpu type LGA1150.