Thanks for posting and welcome to CPCR!
You're right in saying that many gaming systems today still use many of the components you're using in your system. Sure, there are newer processors, but in terms of gaming performance the Corte i5 2500K still keeps up with even Intel's latest Skylake processors. With that in mind, here are my recommendations as far as component upgrades for your system, for gaming.
1. RAM: 4GB is just not enough, and with DDR3 prices so low, there's no reason to not upgrade to 16GB.
2. SSD: Since you only have a 500GB HDD, getting an SSD is definitely worth it as it will improve system speeds drastically, and give you a bit more storage for games. There's also a great deal on the Sandisk Ultra II 480 GB listed below.
3. GPU: For gaming, the most important thing is definitely the GPU, and your aging 560 Ti is fine for older, and less graphically intensive games, but is definitely holding your system back from playing modern, demanding titles. The new R9 380X is definitely the best option under $250, and it will play anything you throw at it at 1080p.
In total the upgrades listed above will run you between $420 and $450, you can shave off about $80 bucks by going with the 240GB SSD, and 8GB of RAM if necessary, but I'd stick with the 380X for the GPU.
You'll also want to consider overclocking your CPU, if you haven't already. Overclocking can make up the difference between newer CPUs and yours, and provide significant framerate increases in demanding titles. You'll need better cooling than the Intel stock cooler, but something like the Cooler Master Hyper 212 will definitely provide sufficient enough cooling to get you into the 4GHz range.