Holmes uses many kinds of evidence to solve the crime in The Hound of the Baskervilles.
1. Dr. Mortimer's testimony: Dr. Mortimer tells Holmes about the tale of the hound and that Sir Charles’s heart had been weak for some time. He also tells Holmes two things he has discovered near Sir Charles’s body: footprints of a huge hound and a cigar that has burned down. Besides, he mentions the change in Sir Charles’s footprints discovered by Barrymore. Based on this information, Holmes deduces that Sir Charles was waiting for someone that night when he saw a huge hound. He was frightened and started to run until his heart suddenly stopped.
2. Strange things that happened to Sir Henry: Two strange things happened to Sir Henry when he was still in London. One is that one of his new shoes was taken and put back; then an old shoe went missing and never showed up. This ensures to Holmes that the hound must be a natural and not a supernatural creature because apparently it needs to smell someone’s belonging in order to track that person. The other thing is that Sir Henry received a letter warning him of the danger in the Baskerville Hall. Holmes detects perfume smell on the letter so he deduces that the letter was written by a woman.
3. Barrymore’s testimony: Barrymore confirmed that Sir Charles was waiting for someone on the night he died. He further told Dr. Watson that Sir Charles received a letter from a woman with initials "L. L.," and he was instructed to burn that letter after reading it. Holmes knows this woman must have a lot to do with Sir Charles’s death.
4. Interview with Mrs. Lyons: Watson visited Mrs. Lyons and confirmed the fact that she was that L.L., who sent the letter and failed to come to the appointment. Holmes learns about her relationship with Mr. Stapleton and deduces that she’s instructed by Mr. Stapleton to send the letter because she wanted to marry him. This was confirmed during the second interview with Mrs. Lyons.
5. Investigation of the Stapletons: Watson reported some strange behaviors of the Stapleton siblings to Holmes. For example, Miss Stapleton mistook Watson for Sir Henry and warned him of the danger. Mr. Stapleton tried to prevent Miss Stapleton from being together with Sir Henry. Learning that the siblings once owned a school in the north of England, Holmes investigates their background and finds that they’re actually husband and wife rather than brother and sister. This makes them very suspicious. Holmes then learns that they come from South America, where Roger Baskerville was supposed to have fled to and died. He also compares the face of Mr. Stapleton with the picture of Hugo Baskerville and finds great resemblance. Therefore, he concludes that Mr. Stapleton is a Baskerville. He murdered Sir Charles and is planning to murder Sir Henry as well so that he can inherit the fortune of the Baskervilles.