Well an i5 would certainly be a great upgrade too, but you'd have to replace your motherboard and probably also your memory (unless you go back a generation to a Haswell processor. Which would hardly be the end of the world; they're still perfectly serviceable CPUs, and a very nice step up from the FX-6300.)
Now if it's just the hardware you need to replace, you could get away with an i5-4460, stock cooler, and also get the PSU, a motherboard which would theoretically work with your current RAM (you would definitely need to check your memory part number to be certain of that!) and an RX 460, which would still be an improvement over the GTX 750 you've got now. (It is a GTX 750, right? Not a GTX 750Ti?) It wouldn't be the quantum leap forward that an RX 470 or 480 would be, though, and if you went that route you'd likely want to look into upgrading the GPU again sooner than if you went with Plan A.
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Core i5-4460 3.2GHz Quad-Core Processor ($174.59 @ OutletPC)
Motherboard: Gigabyte GA-H97M-HD3 Micro ATX LGA1150 Motherboard ($69.99 @ Jet)
Video Card: MSI Radeon RX 460 2GB Video Card ($99.99 @ SuperBiiz)
Power Supply: SeaSonic S12II 620W 80+ Bronze Certified ATX Power Supply ($49.99 @ B&H)
Total: $394.56
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-01-01 02:13 EST-0500
Here's the potential catch, though: a new motherboard might mean you'd need a new Windows license, which would run you another $90. A retail license can be transferred to a new computer (meaning a new motherboard, for practical purposes.) An OEM license can't. Can you find out which one you've got?
Also, what's your case? The motherboard you listed is a Micro ATX model, and so is this one with the i5 here, but it'd be best to check that for compatibility too.
ETA: Newegg's also got an MSI RX-460 for $99.99 that would keep the build under $400 even with the $4 shipping charge, and I prefer them over SuperBiiz. http://www.newegg.com/Product/Product.aspx?Item=N82E16814137028