What does Hamlet believe about his father's death before he meets the ghost?

The ghost is actually the character who reveals to the audience what the whole of Denmark—including his son, Prince Hamlet—believes happened to him. While speaking privately with Hamlet on the castle ramparts, away from the guards and Horatio, the ghost of Hamlet's father says,

'Tis given out that, sleeping in my orchard,A serpent stung me. So the whole ear of DenmarkIs by a forged process of my deathRankly abused. By know, thou noble youth,The serpent that did sting thy father's lifeNow wears his crown (1.5.42–47).

In other words, the story that was shared with the public is that the king was bitten by a snake while he was taking a nap in his garden one day. In reality, however, this is a total fiction, as Claudius, Hamlet's uncle and now stepfather, actually poured poison into his ear canals, and this poison killed him. Old King Hamlet was murdered by his own brother and did not die as a result of a snake bite. The ghost tells Hamlet,

. . . thy uncle stoleWith juice of cursed hebona in a vialAnd in the porches of my ears did pourThe leperous distilment . . . (1.5.68–171).

In this way was he killed, despite the fictionalized story about a snake.

Like just about everyone else in Denmark, Hamlet believes that the death of his father was no more than a tragic accident. The official story is that Hamlet's father, the king, was stung by a poisonous snake one day while sleeping in the garden. However, when the Ghost appears to Hamlet, he tells him what really happened that fateful day: while Hamlet's father was asleep, Claudius crept up to him and poured deadly poison down his ear.
Hamlet hates Claudius right from the start, even before the Ghost appears. He is also deeply resentful of the fact that his mother married Claudius so quickly after his father's death. The shocking revelation gives his hatred and resentment a focus. He swears revenge on Claudius for this act of murderous treachery. Unfortunately, Hamlet, being Hamlet, takes quite some time before finally avenging the death of his father.