Every player as you said is given one free namechange. Here is how you do it:
-Log into your account at the battle.net website.
-Check your registred games and select the sc2 one.
-From what I remember there should be a button to reset your name, once you accepted to do that, start the game and log in.
-It should now be a box asking you to type a new name. Do so and accept, and it should be done.
But yeah, if this is unclear, check the battle net site for info
NOTE
From what I heard, people are experiencing issues with namechanges which only changes the name from capital to small letters and so on. It seems that if you do so the game may screw you over disallowing you to log in at all in the future, unless there already is a fix for this. But I would only change my name if it were to be something new, not to just change the letters to be capitalized or not.
Example: If my name is Mexaprone and I changed it to MeXaPrONe or something like that, it would screw up.