Or you can get an older 4th generation Intel system, keep the RAM if you have 8GB at least, since RAM are expensive.
But even so, your bottleneck caused by the CPU won't be as relevant. Your GPU (I don't know what you're upgrading from, please specify and also your PSU) will still give you the highest boost of power in fps, but if your CPU is really old and not overclocked, you might be below the minimum requirement of speed, which will make your games run very poorly even tho, better.
4th generation CPUs are really really good, especially if you could find a cheap i7 4790k. And you wouldn't have to change everything. Another advice, maybe wait for the new generation of GPUs from Nvidia. Lower prices (especially on used GPUs) or new GPU with better performance.