Anwar al-Awlaki does not consider himself American in the slightest. He considers himself a holy warrior who is, in his words, bound to kill American citizens. He has incited and is inciting Muslims to kill Americans. In other words, a clear and present danger to the US – just like Osama bin Laden.
The People’s View has a great post outlining exactly why President Obama has every right to order US troops to kill this man.