Warning: if you seriously care about spoilers (despite everybody practically knowing the plot of Hamlet already), then don’t read the rest of this. However, I don’t really reveal anything past the first act. I also have not actually read the full play yet; I haven’t even one full act. I do, however, know the plot well-enough to make some educated guesses.
It is clear from the first moment Hamlet is introduced to the audience that serious therapy is in order. First off, his uncle marries his mother. Imagine your father dying and a month later your paternal uncle marries your mother. Secondly, his father (the late King Hamlet) comes back as a ghost and informs Hamlet that his uncle killed King Hamlet. If that was confusing, I kind of apologize, but the gist of it is: Hamlet is sad. Hamlet dislikes stepfather/uncle and takes it out on mother. Hamlet hears ghost of his father is back. Ghost tells him uncle conspired for the throne. And from there, a tale of revenge and ambiguity follows. That doesn’t matter here. If you want to analyze text, take that to the English classroom. Here, we are psychoanalyzing Hamlet’s character.
There are many things amiss with Mr. Hamlet. Hamlet is apparently 30 years old, but throughout most of the acts (until you get to Act V), he appears to be a teenager. He talks back to his newlywed parents, and is so unsure of himself and his decisions, you would think he’s a High School student trying to decide on a coffee order. I mean at 30, you would expect some level of decision-making skills, but apparently not. Also, Hamlet frequently displays some version of the Oedipus Complex. He sexualizes his mother, insulting her for running straight into Claudius’ bed (his now stepfather). Hamlet is a tad disturbed, to say the least.
Most of the time, the audience doesn’t know whether Hamlet is sane. I can honestly not say what types of mental treatment Hamlet should receive, as there are likely many options applicable for him. However, I do recommend some introspection on his part and why he views his mother in such a sexual way.
There are plenty of works that analyze Hamlet’s behavior with much more depth and skill than me, but the gist of it is that Hamlet needs some serious therapy.