It's pushing it to suggest that an FX 6300 has more cores than an i5. It has 3 modules, so 6 integer cores, but each pair of integer cores share a bunch of resources making it often perform more like a 3 core CPU. There are very, very few benchmarks where a "6 core" 6300 will beat an i5 4460 and in most cases the Intel wins solidly even with a hefty OC on the FX chip.
A fairer comparison would be an i3 vs FX 6300/6350. Price wise it's pretty close once you exclude the cooler (not required for an i3).
The FX chips aren't bad for productivity but they still really struggle with gaming. Have a look at the Fallout 4 benchmarks:
http://www.techspot.com/review/1089-fallout-4-benchmarks/page5.html
A (sort of) 8 core FX 9590 @ 4.7Ghz still loses to an i3 at stock. You'd be doing well to get a 6300 to 4.7Ghz with that cooler and motherboard, which would still only approach parity to an i3 using a fraction of the power.
Admittedly that's probably one of the games that puts AMD in a particularly bad light, but still, I question the merits of going FX when for the same price you can get similar performance without the hassle and potential issues associated with an overclock.
i3 + 960 would be my pick.
Back to your original question OP. An i3 is enough for
most games today. There's not many games that show a big difference between an i3 & i5, at least until you have a mid-high-ish end graphics card (like GTX 970 or R9 390 or above). There is the odd game which likes a genuine quad core however, and it's not unreasonable to suspect these will become more common in future. So there's a suggestion that an i5 rig would be relevant for longer. If you're on a tight budget and care about getting the best bang-for-your-buck right now, then the i3 is a good pick. If you have a bigger budget, or prefer to keep the rig for as long as possible, you could make a case to accept slightly lower frame rates right now with an i5 + 950, and then expect that i5 to still be solid and ready for a video card upgrade in 2ish years time. You can make an argument either way... though no doubt that best bang-for-buck right now would be 960 & i3.
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