Since the play begins with a question and is full of philosophical questions, the two major concerns of Hamlet are:
Is man supposed to be an active agent or a passive suffering creature in the world?
Hamlet's famous "To be or not to be" monologue presents two opposing views: "suffer the slings and arrows of outrageous fortune" or "take arms against a sea of troubles"?
Hamlet is not given much of a choice by his father, the Ghost: revenge him and leave Gertrude "to heaven." Hamlet knows he's not cut out to be an avenger, and most of his anger is not directed toward the males; rather, it's focused on his mother. So, how can he summon up the courage to kill a heavily guarded king who may or may not be guilty? It's an existential and profoundly moral dilemma. Hamlet finally decides, "Let be...the readiness is all." He says we must live ready to die.
What is the nature of death and the afterlife?
There's moral confusion in the play and with Hamlet himself. The Ghost's appearance confirms a supernatural afterlife. Hamlet must find a way to kill Claudius and send him to hell, but killing will also send Hamlet to his death and hell as well. There's little chance Hamlet can carry out the Ghost's wishes and both save his life and guarantee his afterlife in heaven.
Also related is the nature of the heavenly father: how can Hamlet's father's Ghost (symbolic of the Old Testament Yahweh) condone revenge if he knows Hamlet is a Christian? So, Hamlet becomes a Christ-figure, torn between the expectations of his vengeful heavenly father and his own beliefs in Christian forgiveness.
There are a number of important themes, obviously, and you can peruse them at your leisure at the rather thorough guide below. But in brief:
Hamlet, the prince, is filled with self doubt and much of the play is centered around his attempts to resolve his own doubts and take action about the fact that he believes his father was murdered. He spends a great deal of time thinking deeply (and telling us all about it) about the various issues, religious, moral, etc. surrounding his currend predicament.
Another major theme is that of deception and intrigue. So much of what happens goes on behind the scenes or involves one character trying to trick or outfox another, or to even keep actions in the dark, that much of the play is about who can figure out what who else has done and how to counteract or explicate that action.
A play that is this famous surely has many themes and concerns. I will discuss a couple of them.
First, you can say that it is a play about revenge and human motivation. Many characters in the play (Hamlet, Laertes, Fortinbras) want to get revenge on someone. We explore their reasons for wanting revenge. We also see that Hamlet is much less motivated than the others and does not seek his revenge as forcefully. Why is this?
Second, this is a play about how unavoidable death is and how it comes about both on purpose and by chance. Polonius is killed by chance. So is Gertrude. Others are killed on purpose. But everyone dies.
At one level, 'Hamlet' is a Revenge play. The ghost of Hamlet's father urges Hamlet to avenge his murder. Revenge seems to be one of the play's main concern / theme.
Another significant theme is that of Madness. There is Hamlet's feigned madness, and Ophelia too lapses into madness before killing herself.
The problematic of evil is a major concern of 'Hamlet'. Claudius, in a conspiratorial intimacy with Hamlet's mother, kills his father to usurp the throne. Hamlet is severely shaken by the pernicious act of his uncle, and more so, of his mother, whom he conceived as the very ideal of a woman. Hamlet's delay in taking the revenge is largely due to his deep sense of evil as integral part of the universe.
We may also look for such concerns / themes in 'Hamlet' as Regicide(the killing of a king), Incest(the relation between Claudius and Gertrude), Death(so many deaths occur in the play) and Misogyny(Hamlet's attitude to woman).