You can update your player name through the in-game store.
To access this service, simply click the "Store" button in the main menu then select "Account" and "Change Name."
The price for a player name change is currently 400 gems.
To change your Player name on Xbox One, you will need to change your gamertag. You can find information on how to do this here.
It is not currently possible to change your player name on PS4, as this is not a service currently offered by Playstation.