Lord Bellinger is the fictional prime minister in this story. We don't see a lot of Lord Bellinger, but he might be loosely based on the actual prime minister. At any rate, Watson and Holmes reveal their patriotic streaks here, and they speak of Lord Bellinger very respectfully. We're surprised Watson didn't ask for his autograph, actually.
Hope is the European Secretary in this book, which is like the equivalent of the Foreign Secretary today, which is the equivalent of the American Secretary of State. Hope committed the ultimate bureaucratic faux pas, or error, by losing a super top secret letter that could possibly start a war if anyone else reads it. Whoops. Luckily for Hope, Lord Bellinger still seems to like him and doesn't fire him for this incident. Hope is worried but he never totally freaks out in the story, which is notable in and of itself.
Lady Hilda is the wife of Trelawney Hope, and she's to blame for this entire mess. Mostly at least. Lady Hilda is kind of foolish when it comes right down to it, though Watson describes her as "noble" for the bulk of the story. Lady Hilda was being blackmailed by a man who apparently belonged to Charles Augustus Milverton's Blackmailers of London clubhouse. So she paid off the blackmailer with some government documents, without bothering to check what exactly she was handing over (a very important letter). Her campaign of "how not to be a sneaky criminal" continues when she goes to chat with Holmes and drops anvil-sized hints about her potential involvement in the whole scheme. Then she steals the important letter back, but fails to return it. And then she refuses to confess everything to Holmes at the end in a very tense scene. She finally caves and spills the beans. Holmes has just enough time to stick the letter back where it belonged. He pretends like it was there the whole time. Lord Bellinger suspects something fishy, but he lets it slide. Lady Hilda gets off thanks to the quick thinking of Holmes. Out of all the women we see in these stories, Lady Hilda is probably the most ridiculous.
Lucas is a spy, and a famous amateur tenor according to a newspaper article that reads like it was written by Watson. Eduardo Lucas actually has a double identity and he spends part of his time in France. The man trades and buys information, but he isn't killed in a spy adventure gone awry. Instead, his wife in France finds out about his secret identity and kills him. So much for being a stellar super spy there, Eduardo. Eduardo is also the man who was blackmailing the short-sighted Lady Hilda. His murder conveniently allows Lady Hilda to steal the secret letter back.
Henri Fourange is the secret identity of Eduardo Lucas and this woman is the wife of that identity. She is French Creole, meaning that some of her ancestors came from French colonies, and she follows her fake-French (or perhaps fake-English) husband back home and kills him in a jealous rage. Interestingly, this isn't the first time a crazy Frenchwoman makes an appearance in English literature. In fact, the crazy Frenchwoman is a standard fall-back villain in a lot of famous books. Dickens actually used this type of character more than once. His A Tale of Two Cities features the sinister Madame Defarge who manages to make knitting creepy. And Bleak House features an angry (and possibly homicidal) French maid named Hortense. Madame Fourange is in good company at any rate.
MacPherson is the officer in charge of guarding the house of Eduardo Lucas, after the place turns into a crime scene. He lets Lady Hilda in to the house after she flirts with him. MacPherson seems like a nice dude though, and he helpfully identifies Lady Hilda from a picture for Sherlock Holmes, which enables Holmes to go confront Hilda about her foolish behavior.
This is Lucas's valet who is arrested for Lucas's murder. He is released after his alibi holds up however.