IT. IS. COMPLICATED.
You can stream without significant CPU performance impact but it depends how you are doing it.
For starters, how is the video ENCODED? If it's for example using the NVidia (NVENC) or AMD hardware encoder on the graphics card rather than a brute-force (CPU intensive) software encoding method then there's minimal to no FPS impact on that alone.
*So experiment with encoding to hard drive to start with.
Assuming you do it that way then STREAMING is basically sending the video out the network rather than to the hard drive.
**How much you can UPLOAD in terms of bandwidth also depends on your ISP upload rate. So no point for example encoding 4K@60FPS if you can't upload beyond a quality 720p@30FPS stream.
Other:
PC?
Having said all that I would not even look at an i7-6700K with the newer Intel CPU's offering six-core variants. If you have a HIGH BUDGET then I'd consider going for an i7-8700K (6C/12T) or i5-8600K (6C/6T) so you future proof any CPU needs.
The i7-6700K is a 4-core CPU... the i7-8700K is a SIX-CORE CPU so there's more than 50% extra processing power available on the newer i7 vs that older one.
Summary:
a) MAY not need much extra CPU
b) Still recommend a good 6-core CPU like the i7-8700K
c) Give a BUDGET so we can use PCPARTPICKER (and state country)
*Graphics card prices are still way over-priced too so if you can hold off consider that... also GSYNC monitors are awesome but expensive (i.e. 27", 2560x1440, IPS, 144Hz, GSYNC is $700USD+).