First of all, this is definitely the older reconstruction of this lost episode. Looking again at the shots of Agamemnon, I can see that they come from Carry on Cleo (though rendered in black and white) and Frances White, who plays Cassandra, is illustrated with shots from I, Claudius, where she played Julia; the shots mostly show Julia either eating figs or yelling at Livia. This partly explains the funny hats - and the extremely comical expression on Priam's face, as some images been taken from Up Pompeii!
The Doctor and Steven finally give up on pretending to be gods or spies and tell the truth, which is so ridiculous that Odysseus decides they can't have made it up. Meanwhile, the TARDIS, with Vicki inside it, has been taken into the city of Troy - which I though was very funny, I must admit. The Fall of Troy was dramatic enough, but imagine what havoc could have been wreaked if the Greek army had had access to the TARDIS... It has been brought in by Paris, wonderfully characterised as a rather ineffectual prince who just wants it all to go away, but who is a decent enough fighter when he needs to be, and who thinks he has captured something valuable to the Greeks.
Cassandra disagrees; she has foreseen the whole Trojan Horse thing in a dream (and thinks it's exactly the sort of thing Odysseus would do) and is therefore suspicious of anything left hanging around outside by the Greeks. Cassandra is one of pop culture's favourite characters from Greek myth, because of the inherant tragedy and pathos in her curse from Apollo - she has the gift of accurate prophecy, but no one will ever believe her and she is doomed to foresee all the horrible things that will happen to her and to Troy, and to be unable to prevent them. She also has a conveniently modern name (or rather, an ancient name that is still in widespread use in English-speaking countries), so she can be referenced in everything from Red Dwarf to Buffy. In this version, her characterisation is a bit different from usual - rather than tragically misunderstood, she is somewhat unpleasant, quick to order the execution of our heroes, and the reason no one will believe her prophecies is because, like Sybil Trelawny (named after another classical prophetess) or Puddleglum, she has a tendency to keep predicting doom and gloom for everyone and they're all fed up of hearing it (of course, the trouble is, she's right).
It's particularly interesting that Cassandra says she foresaw all this in a dream. In Greek myth, Cassandra's gift is simply the ability to know the future - she doesn't need a dream to foresee it, that is the way other people, who are not prophets, catch glimpses of the future. Presumably, this has been altered to try to keep the story 'historical' - many people today would not believe that someone could know the future, but they might suspend disbelief to imagine that someone might have a dream based on known facts (the Trojans love horses, Odysseus will want to smuggle soldiers into the city) that ends up coming true. Tony Keen has talked about the tendency of modern historical fiction to include prophetic dreams on his blog - perhaps this is another example to add. (Or possibly it comes from the later medieval and Shakespearian stories - see below).
The Trojans are about to burn the TARDIS as an offering to the gods, so Vicki is forced to come out and tell them all that she is from the future. Cassandra thinks she means she is a rival prophet and is unimpressed (and incorrectly claims to be an expert in augury - augury refers to a Roman form of divination. She's on safer ground with the general description of divining by the flights of birds or extispicy though). Priam decides to rename Vicki 'Cressida' and suddenly I'm remembering something I read in an old Doctor Who guide about Vicki... Cassandra wants to have her killed as a Greek spy but Priam takes her under his wing. He also orders Paris to go and kill Achilles, to avenge the death of Hector.
Vicki and Barbara, from an earlier and presumably better preserved episode
Back in the Greek camp (which looks like something out of Robin Hood, by the way, presumably another picture taken from a different source), the Doctor has a similar opinion of the whole Trojan Horse incident to me, though he thinks it was an invention of Homer - possible but unlikely, as the Horse only appears in flashback in the Odyssey, and not at all in the Iliad, and was probably an older aspect of the story. Steven seems to be under the (sorely mistaken) impression that the ancient Trojans will abide by the Geneva Convention, and decides to get himself captured so that he can find Vicki, while the Doctor has been given two days to come up with a plan to capture Troy.
This gives a scene that's almost the exact opposite of one of Troy's more memorable setpieces - instead of Achilles yelling for Hector at the top of his voice, we see Paris whispering for Achilles (because he doesn't actually want to be heard, so he won't have to fight). Paris gets Steven instead and agrees to take him prisoner because it gets him out of fighting for the time being.
Back in Troy, 'Cressida' quite fancies Troilus, another prince - which reveals the source for a lot of this to be Shakespeare's Troilus and Cressida and other versions of the Troilus and Cressida story, not Homer or classical myth. That explains a lot! I don't know Troilus and Cressida at all, nor do I know the Chaucer poem or the other medieval versions - but I do know that it is a medieval story with very little basis in classical sources (the Shakespeare play does appear to include material from the Iliad, but the sections relating to Troilus and Cressida themselves are medieval). This makes a brief reference to Diomedes as dead all the more confusing though, since not only does he survive the Iliad, he is instrumental to the plot of the Shakespearian and Chaucerian stories.
At the end of the episode, Paris brings Steven in to Priam and Steven and Vicki recognise each other, leading Cassandra immediately to order both their deaths. Da da daaaaaah!