Well you have 3 options as i see it.
1. overclock the crap out of your current CPU. If properly cooled they can hit 4 GHz+ and with that still be a good CPU for 1080P gaming.
2. replace it with a 6 core model Xeon or I7 model. OBS not all X58 motherboard support Xeon and overclock the crap out of it (i chose to replace my I7 920 with an I7 980X by the way).
3. sell your old CPU, mobo and memory and kill your piggybank and bay new CPU, mobo, DDR4 ram and cpu cooler.
But if a new CPU why a I7 6700K and not the latest cpu from intel that is I7 7700K?
If at any interest here are my old I7 920 system and my current I7 980X system for a comparison.
OBS: for every day use my I7 920 ran 4.1 GHz and not 4.4 as shown here and my I7 980X is clokket in at 4.25 GHz now.
http://peecee.dk/uploads/032017/3dmark_firestrike_rekord_1.jpg
http://peecee.dk/uploads/032017/3Dmark_Fire_Strike_I7_980X_4.3_GHz.jpg