The difference between i7 3930k and i7 6700k does not worth the price you have to pay, unless you are feaking rich.
Just keep in mind, i7 6700k does already noticeable faster than i7 3930k. This is already a sign for you that you would probably have to upgrade your i7 3930k within the next 1-2 generation after Skylake. But for now, just OC it.
Beside OC,
For streaming games, removing other bottlenecks is also important...so....make sure, you have at least 16GB RAM and an SSD. Plan to get them, if you do not have them yet.