What is the real and unreal story about Hamlet's father death?

I'm not sure what you mean, are you talking about the facts as the ghost presents them? Or Claudius and Gertrude's version.

The facts as the ghost presents them

It's all right there in the first part of Act 1 Scene 5. The Ghost tells Hamlet something to the effect -- Tis given out that sleeping in my orchard a serpent stung me so the whole ear of Denmark is...rankly abused. -- In other words the public "story" of the King's death is that a snake bit him and he died. But the Ghost then explains what really killed him. Claudius, his brother, snuck up on him with a vial of hebanon, a poison, and Claudius poured this stuff into the sleeping king's ear and killed him. As the ghost says the serpent that stung him now wears his crown.