CAT6 will be an overkill and useless.
Your Xbox One is having A Gigabit NIC (1000Mbps). Cat5e supports up to 1000Mbps.
Using a CAT6 can exceed speed above 1000Mbps, but because your Xbox One doesn't support that, it is useless to CAT6.
Also make sure your computer network card is a Gigabit network card, otherwise you will only get 100Mbps.
So a direct answer to you:
Using CAT6 as network cable between your computer and your Xbox One will not increase speeds, unless you are using a CAT5 network cable now.
CAT5e is the standard network cable for the last 2 years, slowly we are moving to CAT6, but for now CAT6 is mostly used in industrial environments. CAT6 is mainly used for networks that have 10GBASE-T switches etc.
Well you can use CAT6 there is not much difference in price between CAT5e, and CAT6 (about 10-20% difference), but do not expect to see much improvements in speed.