This is done on your User Profile, which by clicking the User icon at the top right of any Gamer Launch site you're logged into, then clicking your username from the dropdown. 

When you're on your User Profile, follow these steps:

1. Click Edit Profile on the left column.
   
2. Change the field you'd like to modify (whether Password, Display Name or email address).
3. Click Update.
   

That's all there is to it!  Your username has been changed.  You will still need to choose a unique handle; if it is taken, then you will need to pick another username (but fear not - there is still a way to post as your gaming handle/the character your guild mates know you by!  Check out this article for more details!)