There are many scenes in Hamlet you could choose and use to justify your claim that Hamlet experiences a reversal of fortune. Personally, I would choose the following: (1) the ghost's request that Hamlet avenge his father's murder in Act I, Scene 5 (2) Ophelia obeying Polonius's orders to break up with Hamlet in Act III, Scene 1 and (3) Hamlet's killing of Polonius in Act III, Scene 4.
We understand what Hamlet's life might have been from Ophelia's soliloquy:
O, what a noble mind is here overthrown!
The courtier's, soldier's scholar's, eye, tongue, sword,
H'expectancy and rose of the fair state... (Act III, Scene 1, lines 150-152)
Before the play begins, Hamlet was the golden prince —noble, learned, good-looking, and potential heir to the throne. When the play begins, Hamlet is grieving his father's death and is disappointed in his mother's hasty marriage to Claudius. While troubling, these are developments Hamlet might have overcome, but his confrontation with his father's ghost at the end of Act 1 sets him on an entirely different course. Hamlet knows his purpose must be to avenge his father's death by his uncle, and this revenge will likely change his life forever. To exact revenge, Hamlet must become a killer himself, something totally against his nature. Hamlet feels the weight of the ghost's command upon him when he says,
The time is out of joint. O cursed spite
That ever I was born to set it right (Act I, Scene 5, lines 190-191).
The second scene I would chose is the one in which Ophelia breaks up with Hamlet. By agreeing to do break up with Hamlet, Ophelia has in essence chosen her obedience to her father over her love for Hamlet. Hamlet, already without his father and unable to confide in his mother because she married Claudius, now loses another ally. It is in this scene that Hamlet realizes how alone he really is in his revenge. Several times during this scene, he tells Ophelia to go "to a nunnery." Hamlet is ending his relationship with Ophelia forever. With the exception of Horatio, Hamlet has now lost his entire support system.
The scene that brings Hamlet's plot of revenge to a screeching halt is his murder of Polonius. This failed attempt to kill who he thinks is Claudius exposes Hamlet's intentions to Claudius and puts him on the run. It also causes Ophelia's mental breakdown and makes a passionate enemy of Laertes, who now wants "to cut his throat in the church" (Act IV, Scene 7, line 123)!
In Act III, Scene 4, Hamlet tells his mother that he must run and hope to evade Claudius's attempt to have him killed. Claudius and Hamlet have now reversed roles; Claudius wants to kill Hamlet.