The 6700K would be a far better option, as would a 5820K, but I assumed you might've been considering an old and discontinued platform so you could reuse your RAM - the 6700K and 5820K both use DDR4.
Honestly, in gaming, you probably won't see much, if any difference, because the 4770 is such a high-end CPU to begin with and few games are likely CPU-bound with what you have. You'll probably see a more noticeable improvement in editing with a 5820K than a 6700K, due to the extra cores.