Changing your name in game:
If you haven't already set your player name, you can change it to match your Kongregate.com name by clicking the Red K in game and signing in.
This will change your Player-1234 name to whatever your Kongregate name is.
Aside from this, there is no way to change your in game name yourself.
Changing your Kongregate.com name:
The option to change your Kongregate.com name is only available if you have registered for Kongregate through Facebook.
This is because Facebook automatically assigns you a name that some may feel is personally identifiable. The option for these accounts will stay open for a few months, but eventually expires and that person can no longer change it.
Changing your Kongregate name will also change your name in Stormbound.
If there is an issue with your name such as it being personally identifiable, you can request a change through Kongregate support at http://kongregate.com/feedbacks/new.