D'oh.

Enter the character's inventory, there's an icon inside the character's portrait that let's you do it.

On all the NPC's there's a door icon on their portrait - just click on that.

For player characters, click on their name (the same way you would do change their portrait or name) and there's an icon at the lower left screen that lets you delete them - then you can choose stats, etc. for a new character and choose whether to buy equipment for them. I wouldn't recommend doing this unless you're still early in the game, because the new character starts from scratch. ;)