Claudius - Derek Jacobi
Augustus - Brian Blessed
Livia - Sian Phillips
"I, Claudius" is one of my all-time favorite TV shows, and my absolute favorite with a disabled character.
Even though "Grandmother" Livia is nasty pretty much to everyone, her insults and eye-rolling attitude towards Claudius are just somewhat extreme versions of what most people around Claudius think of him. He's a "halfwit" who's embarrassing and annoying to have around. This scene also shows how peoples' impatience with Claudius' disabilities make the disabilities more pronounced; the more Livia sighs and calls him an embarrassment, the more Claudius stutters and the clumsier he becomes.
A major theme of the story is that Claudius' "infirmities" actually save him from most of the intrigues and literal backstabbing that plagues his family. When Emperor Caligula is assassinated, the Preatorian Guard, grab Claudius from where he's hiding, and instead of killing him as they're doing to the rest of the imperial family, laughingly declare him to be the new Emperor. It is implied that they, too, see Claudius as a halfwit … not responsible for Caligula's horrible behaviors … and someone they will be able to control. As both the story and actual history make clear, they made a pretty good choice, but not for the reasons they thought.
In addition to being a key turning point in the larger story, Claudius' exchange with the Senators is an eloquent statement of the contrast between specific impairments and a person's actual worth.