How do you delete a character in starbound?

Go to Steam\SteamApps\common\Starbound\player. There you find your character and world files. At a first glance, there is no way to tell which one is which though, since it uses some sort of hasing to create filenames. So you should look at the file creation date and also make a backup before deleting it.

Alternativly, you can open the file with notepad and have a peek inside. Your name is there on the top after the "ACK" Symbol:

My character name is Gordon, so I know this is the right file.

Upbeat Giraffe has added delete capability to the character select screen. Simply click on the trash icon associated with the character and you will be able to delete it.