Hamlet is undoubtedly one of Shakespeare's most famous and widely performed (both on stage and in film) tragedies. What makes the play unique compared to his other works, I feel, is the way he uses Hamlet's introspection to drive the plot of the story. Rather than focusing on events unfolding around the characters, Shakespeare creates a character who engages in constant, intense processing of his own thoughts and emotions.
Most of the play focuses almost exclusively on Hamlet's own psychological struggles and emotional torment. I would certainly call the inner torture Hamlet experiences disturbing; he is plagued by visions of his father, who was murdered by his uncle, and becomes increasingly volatile and unpredictable as the play progresses. The overall tone is dark, foreboding, and frankly, quite spooky.
Now, to address the "psychological thriller" aspect of the question, I would recall how open to interpretation Shakespeare has left this play. We can question the motivations and actions of all the characters, not just Hamlet and his emotional state. Take the famously ambiguous scene of poor Ophelia's death, for example (see act 4, scene 7). The Queen claims to have witnessed the entire thing, describing it in dramatic detail to Laertes. Why didn't she help? Why was she there? Could she have played a part in Ophelia's death? That is only one snippet of the play, but it creates a bit of a murder mystery in itself.
I think it's certainly fair to characterize Hamlet as a disturbing psychological thriller. If you're writing an essay to tackle agreeing with the statement, consider the parts of the play that really left you questioning what was going to happen next. Make note of the times you questioned a character's true motivations or speculated about what really happened. Hamlet is a very unpredictable, ambiguous play; you can interpret almost every aspect of it, which might be what Shakespeare intended.