Officially the Black Ops 3 Beta for Xbox One starts on August 26th. We are hoping that it will start earlier than this, the beta for the Playstation 4 started a day early.
In order to have access to the Black Ops 3 beta on the Xbox One you need to have pre-ordered the game but there may be a chance to play it later into the beta test similar to what happened for PS4 users.
There is no official confirmation on the early start but hopefully Activision / Treyarch will send out the codes early.
The size is between 12GB-15GB so make sure you have a fast internet connection and space on your console otherwise you will waste a day of the beta.
Expect a few problems, this is a beta.