"Conspiracy" has an extremely tricky tightrope to walk. On the one hand, it's a sex-comedy farce for a good portion of its running time, with Nero and Barbara running about the palace and almost bumping into each other and everybody else as Nero engages in an extended "aren't I a naughty boy?" romp, with Barbara as his intended conquest. On the other hand, "conquest" isn't really the right word. "Victim" is really more accurate, here--for the tension that drives rest of the script to work, it has to be very clear that Barbara has been caught in between two of the most dangerous people in Rome, and she takes her life into her own hands whether she refuses Nero's advances or not. (And the tightrope is even tighter when you take into account the change in society's attitude towards sexual harassment and sexual assault over the years--the episode wouldn't have aged nearly as well if they'd played Barbara's sexual peril purely for laughs.)
For the most part, the episode succeeds by making it clear that Nero genuinely is a monster. He's a monster who would be laughable if he wasn't the Emperor, of course; his self-pity, rampant egotism and comic ineptitude is a thing of beauty, especially with Derek Francis chewing exactly the right amount of scenery for the part. But if he's a narcissistic idiot, he's a narcissistic idiot with the power of life and death over everyone else in the story, and that makes him simultaneously hilarious and terrifying. He may be chasing Barbara around the bed like he's in a sub-par Benny Hill skit, but there's no question that if he ever grows tired of chasing her, he can have her put to the sword and nobody will say a word. The death of Tigilinus shows just how thin the line between comedy and drama is, here--after a full episode of narrowly avoiding Nero's wrath, he's killed simply to prove a point.
Everything else in the episode works because it walks the same tightrope. Locusta, the court poisoner, goes about her business with the same matter-of-fact disinterest as an undertaker (albeit a bit earlier in the process). Nonetheless, she's condemned to death for failing Poppaea. Vicki is cautioned not to change history, and responds with a decision that leads to what has to be the best line in the episode: "Oh, by the way, I think I poisoned Emperor Nero." The Doctor manipulates all of Nero's very obvious levers with brilliant aplomb (and it's a treat to see Hartnell and Francis playing off each other in the sauna scene, it really is) but winds up infuriating the Emperor nonetheless. At every stage and every step, comedy and drama perform an uneasy dance together...but the episode rarely puts a foot wrong.
Ian, unfortunately, doesn't get to participate in any of the comedy. His plot is basically a tour of the brutal Roman tropes, so he goes from galley slave to gladiator. His subsequent fight with his new friend Delos is as inevitable as water running downhill, right down to the cliffhanger of the episode. If you put a gladiator fight into a Roman story, is there really any way it doesn't end with the emperor turning his thumb in disapproval?