They weren't Americans, and there is no war on terror, because there is no war.
There is the Authorized Use of Military Force, passed by Congress on September 14, 2001. In that legislation, in essence the Congress authorized the President to go after Al Qaeda, to prevent future terrorist attacks on the U.S. You are correct that it is not technically a war, but it isn't the President acting on his own.
If Obama had actually followed procedure, he'd have said he did so in an effort to reduce the condemnation he received for murdering an American citizen with a drone.
The Administration thinks they did use the process that Congress allowed:
"We believe that the AUMF's authority to use lethal force abroad also may apply in appropriate circumstances to a United States citizen who is part of the forces of an enemy authorization within the scope of the force authorization," reads the Justice Department memorandum, written for attorney general Eric Holder on 16 July 2010 and ostensibly intended strictly for Awlaki's case.
Among those circumstances: "Where high-level government officials have determined that a capture operation is infeasible and that the targeted person is part of a dangerous enemy force and is engaged in activities that pose a continued and imminent threat to US persons or interests."
Basically, Congress had authorized use of military force on Al Qaeda, so they used military force on a member of Al Qaeda. The AUMF makes no distinctions of national allegiance, and doesn't require a court clear every offensive action.