We also see through many novels (and onscreen) that one's House is often more important than one's actual skill (which explains some things about the Empire, really). Alexander isn't just the son of the previous ambassador (and of an earlier ambassador), he's also a member of Martok's House.I never thought Klingons went by qualifications when they hire leaders. Lets see to become captain you basically fight to the death with the people who are a higher rank.
I'd wager that the Federation would've accepted his appointment based on that (knowing that Martok is practically guaranteed to listen to what Alexander has to say). The fact that he was raised for much of his life aboard a Starfleet ship and already has dual-citizenship is just gravy.
We know they found him entertaining and harmless if not always useful. Which is probably how much of the Empire views the Federation, come to that. . .Lastly we don't know what his military record is, and since he isn't dead yet his shipmates must of thought he was usefull.
"Diplomat" being the Klingon word for "lousy fighter?"Knowing Klingons, losing a fight is probably what gets you an ambassadorship.
