What historical source might Shakespeare be using as a reference for Hamlet?

Shakespeare's source for Hamlet is generally accept to have been Thomas Kyd's translation of an old Norse legend that first appeared in the French language in Paris in the 1500s, later appearing in English in 1608. No extant copies exist of Kyd's translation of the Norse tale, but it is called Ur-Hamlet ("ur" means "original). Very little is known of Kyd's work except that it was a tragedy that was performed in London and a character named Hamlet saw a ghost that cried "Hamlet, revenge!"

The old Norse legend was written by Saxo Grammaticus and told the story of the rulers of Denmark. It was called Gesta Danorum, which translates as History of the Danes. The story about the Hamlet prototype is in Books 3 and 4. In Saxo's legend, the King is Rorik. He marries Garuth and they have a son called Amleth. Rorik's brother kills him so he can marry Garuth and become King. Amleth fears for his life and pretends to be mad. He plans revenge, which succeeds, and he becomes the rightful King.