It is certainly plausible that Hamlet would have deduced that his father had been assassinated had the ghost of King Hamlet not appeared to him. For, as a sensitive and deliberate young man with a tremendous power of feeling, surely Hamlet would have detected suspicious behaviors on the part of Claudius:
- Claudius has quickly married Gertrude after the death of his brother, King Hamlet.
In his first and second soliloquies, Hamlet bemoans his state in life with the death of his father
So excellent a king,...
Hyperion to a satyr, so loving to my mother (1.2.139-140)
and is angered over the "frailty of his mother, who within a month has married his uncle, "O most pernicious woman!" (1.5.105); moreover, he feels great antipathy towards Claudius,
O villain, villain, smiling, damned villain!....
That one may smile, and smile, and be a villain.
At least I am sure it may be so in Denmark.
So, uncle, there you are....(1.5.106-109)
These strong feelings and Hamlet's perspicacious nature would lead him to be cautious around Claudius and, certainly, suspicious of him.
- Claudius plans to send Hamlet to England and distance his nephew from the court in Denmark.
While Polonius believes that Hamlet's "madness" stems from his relationship with Ophelia, after eavesdropping with Polonius upon Hamlet and Ophelia, Claudius's conscience leads him to think that Hamlet has secrets which may prove dangerous to him:
...There's something in his soul
O'er which his melancholy sits on brood,
And I do doubt the hatch and the disclose
Will be some danger....(3.1.160-163)
Therefore, he plans to send Hamlet away, putting "seas" between Hamlet and Denmark. Whether he would have gone through with his plan to have Hamlet executed in England is debatable because the "dumbshow" performed by the visiting players is most responsible for raising his suspicions, and Hamlet would not have directed such a show if there were no ghost of his father. Nevertheless, Claudius still seems overly fearful of Hamlet as he feels that Hamlet is like a fever in his blood--"For like the hectic in my blood he rages" (4.3.62).
- Claudius's multiple efforts to deflect attention upon himself implicate the king's nefarious nature.
His machinations to implicate Hamlet at every turn certainly raise suspicions, and eventually entangle him. When Laertes desires to kill Hamlet for the murder of his father, Claudius becomes entangled in this vendetta and, in so doing, becomes suspect of some ill-doing. Certainly, Hamlet recognizes Claudius's ulterior motives in this duel between him and Laertes as Claudius has the poisoned wine.