The new 6xxx series from Intel also requires new DDR4 RAM which is expensive, and the PSU would need to be addressed (need the right connectors and enough power for the Mobo for a 6xxx series which is different than any series before), and most of all IF your current PC is a OEM (i.e. Dell, Gateway,etc.) you can NOT swap the CPU / Mobo out as Windows license detects this and blocks Windows from running, installing, repairing, etc. until you go back to the hardware you were originally sold. You would need a new copy of Windows 'off the shelf' (aka Walmart, Amazon,etc.) and WIPE and CLEAN INSTALL FROM SCRATCH Windows and all your programs, games, and data.
If your really a tight 'budget' (usually means people trying to stick under $400US) than buy a console (PS4) to get the best BANG for BUCK in performance of cost as you can play HIGH graphics (1080P) while at 30FPS to 60FPS on almost ALL titles (depends on the game). Otherwise you would be focused on $750+ costs for a PC gaming system, usually with a i5 (would stick with the 5xxx series) and say a GTX 970 as the best Bang for the buck. I personally just had to build / buy my system, and settled on a Xeon with GTX 970 new Mobo, 16GB RAM, etc. totalling just over $1K US.