Furious, apoplectic, absolutely beside oneself with rage. (Imagine a rabid dog, foaming at the mouth and - wait for it! - barking...)
Can also refer to someone who is unbalanced (using "mad" in the British sense, referring to an insane person).
I think Dick Cheney's just barking mad.
(UK Usage) - Almost Insane, usually in a socially acceptable manner. Often used to describe the behaviour expected of some (rural) old money families and more a likable eccentricity than a true mental disfunction.
By god, their all barking mad, colonel!... See if they'd like to come for supper next week when we have the Hybrid-Smyths over. I think they'll all get on famously.