Augustus attempting to use Claudius as an example of a 'good Roman boy' early in the series.
Caligula's performance as Dawn must be seen to be believed. Especially funny when you consider that John Hurt crafted his own getup for the scene. Especially his audience's reactions. One wonders how many takes they needed to film before Derek Jacobi and the others stopped bursting out laughing.
Augustus going around the block talking to his daughter's numerous lovers. "Ah, only once. That's all?"
When Tiberius receives news that Lucius has died and he is summoned back to Rome, he and his astrologer (whom he's promised to kill if there was no good news) both burst out laughing. And laughing, and laughing, and laughing, while the messenger just stands there uncomfortably and continues to tell the tragic details of the story while they keep laughing.
Messenger: Sir, all Rome is drowned in grief! Tiberius: Well, of course they are. It's only natural. Ha, ha, ha, ha, ha, ha!
Later, he excuses it as "nervous laughter" and pretends to wipe a tear. Then he starts laughing again.
Tiberius, about his mother: "They say a snake bit her once, and died."
Everyone's initial reactions to Caligula's announcement that he's Zeus.
Claudius: He wants to see you; he's become a god. Oh, you're a god too. (To Herod) We're not.
There is a dark sort of humour in the scene where Claudius chats to Caligula's wife about all the horrible depravities Caligula is committing while making kootchy-koo faces at her baby.
Antonia: Sejanus will allow it, I'm sure. He has nothing but contempt for you anyway. He'll suspect nothing. Claudius: (beat) Thank you.
When a pair of rival historians learn Claudius has read both their works and demand to know who he prefers, he says that it depends on what he's reading for. He'll read one of them if he wants beautiful language...and the other if he wants to know what actually happened. Naturally, the first one is quite upset at this, though Claudius genuinely thought he was giving a compliment.
Caligula's casual statement that he doesn't know Macro, but he's slept with his wife several times.
When Scilla, one of Rome's greatest prostitutes, learns that she's expected to compete with Messalina in a competition to see who can bed the most men when there's no money in it, she's not impressed. Especially when one of the assembled group of Messalina's hangers-on chides her for actually wanting to be paid, rather than raring for the chance to defend her reputation. She retorts with this gem:
"The difference between you and me, actor, is you're a snob and I'm not. And the difference between this great lady and myself, is that my work is her hobby. My hobby happens to be gardening, for which I don't expect to be paid."
The way the Praetorian Guard, upon the death of Caligula, realize they can keep their cushy jobs by nominating Claudius as their emperor. The scene alternates between funny and tragic as poor Claudius keeps trying to tell them "No! I wh-wh-want a Republic!"
Sergeant: Now, now, sir, not in front of the Germans...