Some of the clues are: the aunt and mother are twins, Rachett is Edmund's stepfather, the stolen painting, finding the buttons in the cemetery and the vault, the opening in the vault being too small for an adult, that aunty was not wearing her own dress and what Sunrise refers to. There are many other small clues throughout the book but these will build to the solution.
The mystery is solved by Poe (Dupin). He unravels the events for Edmund then refuses to help him. Poe has the information he needs to write his story and Edmund must continue on his own. He uses the information Poe gave him to find his mother and discovers that she came to America to get her money and a divorce from Rachett. Rachett, afraid of being discovered for the man he is, kidnaps Edmund's mother. She is able to get a message to her sister who brings the children to America to rescue her. Rachett has a man he is working with, Peterson and they came up with plan. They want to kidnap the children as well and get rid of anyone, the aunt, that does not fall into their scheme. They use Sis to steal the gold and plan to leave with it but Poe and Edmund got in the way.