pieced together into a picture untilthe Ghost gave him the one missing piece of the puzzle. For example, his uncle and mother were showing unusual concern about what he was thinking and feeling. Claudius was acting much differently than his uncle had acted in the past, and the new king was doing an unusual amount of drinking. The story about a serpent killing Hamlet's father was fishy enough in itself. Such a thing might happen in India or Borneo, but how often do poisonous snakes kill people in a cold climate like Denmark's? In fact, are there any poisonous snakes in Denmark at all? 

Hamlet is sincerely overwhelmed with grief at the death of his father. He is also disgusted with his mother. He has plenty of things to occupy his mind without focusing on his feelings, or suspicions, or vague intuitions, about Claudius. Hamlet might have picked up clues long before his father's death that would make him sense that his uncle might be sexually attracted to Gertrude and might have sinister ambitions. Our unconscious minds will often give us warnings in our dreams. It might be said that we all have "prophetic souls," but most of us often fail to heed them. This, in fact, was true of Hamlet. He was all wrapped up in his studies of languages, ancient history, and philosophy at Wittenberg and didn't pay attention to practical matters at home. Otherwise, he might have become king instead of his wicked and cunning uncle.

Hamlet has been uneasy about Claudius' assention to the throne and his over-hasty marriage to Gertrude.  Hamlet has a prophetic soul to the extent that he has not liked Claudius from the start of the play, so hearing that Claudius murdered King Hamlet was not that big of a surprise.   I don't think he suspected him of murder -- no one suspected murder.  The ghost says that the story was out that serpent bit Claudius in the ear while he was sleeping in the garden.  I don't think there was any thing else suspected by anyone in the court.  There is no mention of an investigation or anything like that.  Claudius has control of the court as we see in his first monologue in Act 1 Scene 2. 

Hamlet meets the ghost for the first time in this scene, and the ghost tells Hamlet that Claudius, Hamlet's uncle, killed Hamlet's father. Hamlet's response to finding out that his uncle killed his father is, "O my prophetic soul!/Mine uncle!". Hamlet had suspected that Claudius had something to do with his father's death all along. So when he refers to his "prophetic soul", he means that deep down inside, he knew his uncle was guilty of the murder of his father. "Prophetic" means predictive of something to come, and Hamlet's "soul", the very essence of him, was trying to tell him that Claudius is guilty. The soul is personified since it's given the human action of prophecy.

Since a prophecy is a prediction, we first need to attend to what predictions Hamlet had made prior to this scene.

In act 1, scene 2, Hamlet stands on stage giving a long aside. In this speech, he talks about the speedy marriage of his mother to Claudius and how something just doesn't seem right about the union. He remembers how well his father loved his mother: "So loving to my mother / That he might not beteem the winds of heaven / Visit her face too roughly."

But his mother has married his father's brother within a month, and Hamlet considers his uncle no more similar to his father than he (Hamlet) is to Hercules. This marriage has distressed him, and he remains suspicious of the underlying motives: "It is not, nor it cannot come to, good." This is the first of the prophecies of Hamlet's soul.

Later, in act 1, scene 2, Hamlet has been conversing with Horatio, Marcellus, and Bernardo, who have told him that they have seen what appears to be his father's ghost. Hamlet wishes to speak to the apparition and comments, "My father's spirit in arms! All is not well. I doubt some foul play." This is the second of Hamlet's prophecies.

When Hamlet speaks with his father's ghost in act 1, scene 5, he is referring to these earlier asides where he knew that his mother's marriage to his uncle was suspicious and that there was foul play involved. This new knowledge (which his soul had perceived all along) drives the primary conflict in the play.

Hamlet says this right after the ghost explains that he was not bitten by a poisonous snake but poisoned by Claudius. What Hamlet means is 'I knew something was wrong' or 'I had a gut instinct that some evil lay in my uncle's heart.' Hamlet, of course, had no idea exactly what the evil was, which is why he reacts with such shock to the ghost's detailed expose of events.

Interestingly, although the ghost confirms what Hamlet suspected—that something was wrong about how his uncle became king—Hamlet does not immediately rush to avenge his father's murder. Instead, his head rules, and he moves step by step to gain solid evidence that will confirm that the words of the ghost are real. He doesn't want what we today might call confirmation bias, which would be believing the ghost's words because the ghost tells him what he wants to hear, to influence his decision to seek revenge.

This quote suggests that Hamlet has thought all alone that his uncle was somehow responsible for his father's death. His soliloquies have made it clear that he is very unhappy with his father's death and his mother's rapid remarriage. In fact, it is pretty clear that he hates his uncle, claiming that he is a "satyr" to his father's "Hyperion." But he has not explicitly said he suspects his uncle of murder. But his father's story of his murder obviously touches a nerve with Hamlet, who seems to have recognized that something was not right about his father's death, even beyond the fact of his mother's rapid remarriage.