Intel will be releasing the Skylake CPU's in August, if you can wait until then, you may get a better deal. If you have the extra cash on hand, you can consider the GIGABYTE GV-N960IXOC-2GD GeForce GTX 960 2GB which is a bit more powerful than a 750 Ti. If you replace the motherboard, you will need to repurchase a copy of Windows to remain compliant with the license if you are using an OEM/System Builder copy of the OS. Microsoft loves to change things around with licenses for System Builders, you can read about it here.
Use caution when looking at benchmarks, the best metric to look at when evaluating performance is actual real-life actions and not a synthetic battery of tests. With the FX-6300 series, you may get better multithreaded performance especially if more threads are thrown at it, however, for lightly threaded programs, the higher IPC (Instructions per cycle) of the i3 would give it the performance edge. AMD will be releasing their Zen CPU's eventually, but like the Skylake models Intel is releasing, these use a different socket needing a different motherboard. AMD tends to keep sockets around for a long time.
Your current motherboard can handle the high-end AMD processors. You can save some money by getting one of the FX-8300 series CPU's. I believe this would be a more cost-effective option. I would go with a AMD FX-8320 (Assuming this is your motherboard, then yes, it is compatible), a CORSAIR CSM Series CS550M 550W power supply, and a GIGABYTE GV-N960IXOC-2GD GeForce GTX 960 2GB GPU. If you have extra money, you could even get a more powerful GPU like the ZOTAC ZT-90101-10P GeForce GTX 970 4GB. This build with the 960 is more powerful and you won't need to rebuy and reinstall Windows. As for a new heatsink, assuming your case can handle it, the Cooler Master Hyper 212 Plus gets good reviews. These big heatsinks sometimes have issues clearing RAM if they have tall heat spreaders on them.