Cat5e is gigabit not sure how much more you need because you won't get a NIC to go above that unless you go to fibre.
Cat6 is very particular to cable lengths and patch lengths in particular can be problematic and cause interesting losses that aren't as apparent in Cat5e.
In my past life all my servers were fibre connected with fibre campus backbone and gigabit to the desktop via Cisco 6500 Catalyst switches.
All the desktops had 10/100 NICs which supported fully converged voice/data/video. (The voice and video was my domain)
I did wire the services in my sons house in Cat6 because I got it for $0.00
Damn I happy to be retired and don't have to do this stuff anymore.
Edited to add. Don't let that put anyone off using Cat6 just be aware it can be difficult.