Cyril is one of the funniest characters featured in these stories. A rugby player, Cyril is basically a stereotypical jock in a lot of ways. He's loud, he uses lots of slang, he doesn't express himself all that clearly (though Watson notes that he's cleaned up Cyril's narrative for us). Cyril barges in on Holmes and is hilariously confused by the fact that Holmes knows nothing about rugby and has no idea who Cyril is. Overall, Cyril seems like a nice guy and he is genuinely concerned for his friend and teammate, Godfrey Staunton. His concern is also that of a friend, not just an athlete who is worried his team will lose their game without their star player.
Godfrey is the star rugby player who disappears mysteriously before a big game. We never actually hear Godfrey speak in this story, and Holmes himself never has any direct contact with the man. The reason for this is because Godfrey disappeared to be with his wife on her deathbed. When Holmes and Watson finally locate Godfrey, he is crying over his wife's very recent death. The two leave him in privacy after this. Aside from being a very tragic figure, Staunton also shows us how strict and difficult class status was in this era. Staunton is the nephew of a wealthy man with very rigid ideas of propriety. As a result, Staunton had to keep his marriage a secret, since he didn't want to risk getting disowned by his wealthy uncle for marrying a relatively poor woman.
The Lord is a bit of a Silas Marner/Ebenezer Scrooge character, though he isn't around long enough to have a reformation story line. Lord Mount-James is a miser, meaning that he's stingy with his money. He's a bit like the Duke in the "Priory School" in that he cares more about money and potential scandals than his relatives. Holmes doesn't care much for people like that, and he plays the Lord accordingly in this story. Holmes suggests to the Lord that if Godfrey were kidnapped he would be asked for ransom. At the thought of losing money, the Lord panics and agrees to pay Holmes to find Godfrey before a ransom note is issued. The Lord's ridiculous attitude is also why Godfrey felt the need to marry in secret.
Dr. Armstrong is one of the very few intellectual rivals that Holmes encounters in these stories. Armstrong also stands out for the fact that he isn't an evil super-villain. Really, he's more of a foil than anything else. Both Holmes and Armstrong are working for good, but they are at cross purposes for much of the story. Armstrong is one of the few people to successfully elude Holmes for a period and he nearly manages to throw Holmes off the scent of the missing Godfrey (until Holmes enlists and actual dog to help him find the trail). In the end, Armstrong and Holmes realize they both have Godfrey's best interests in mind, and the two part in mutual respect.
These are the names of rugby players that Cyril Overton references when he first meets Holmes and Watson. By dropping names like this, it's apparent that Cyril expects his audience to know who these people are. Holmes is clueless though since he doesn't follow rugby, which Cyril finds incomprehensible.
We only hear about these individuals second hand for much of the story, and Holmes and Watson only see Mrs. Staunton moments after she died from an illness under Dr. Armstrong's care. In typical Watson fashion, he describes Mrs. Staunton in almost angelic terms. Her father seems like a nice person, and he appears to have a decent relationship with his son-in-law, Godfrey, despite the secret nature of the marriage.
Pompey is the dog belonging to a guy named Jeremy Dixon. Holmes hires Pompey for the day to help him track down Dr. Armstrong and Godfrey and Pompey does so successfully. This isn't the first time that Holmes has used dogs to help him solve a case, and he appears to be fond of them. Another notable case where Holmes used a dog was in the novella "The Sign of Four," one of the earliest Holmes stories.