How can I change my Minecraft username?

Simply said, this is not possible. As stated on the official Mojang site:

Changing usernames

We do not currently support changing, editing, or correcting usernames, though this may be a feature in the future. Make sure that you choose wisely; it will be used for your multiplayer persona and is visible to others.

However, Mojang is working on a way to make this possible in the future using the new Mojang accounts system. For now though, you can't change it.