Hamlet's external conflict develops because of his uncle and mother wanting him to stop moping about over his father's death. Also, there are the conflicts brought to the fore by Fortinbras and his moves upon Denmark. He also has conflict because of the marriage of his mother to his uncle. Another external conflict is that he has been asked not to go back to university. Also, he encounters external conflicts that his own actions bring down upon himself and others: his murder of Polonius, Ophelia's rejection of his love, the fatal duel with Laertes.
Most of the conflict in the play is a result of the internal conflict that Hamlet has over his repugnance with his mother's marriage, his knowledge's of his uncle's treason, and his hesitation to go against his religious beliefs and his mother in the face of his quest for revenge.
Internally, he despises his mother for getting married so fast to the brother who murdered his father. Once the Ghost informs him of the murder, for the rest of the play, Hamlet struggles internally between wanting more sound proof of Claudius's guilt so as to avoid regicide and his desire to kill him. This internal conflict leads to a lot of the external conflicts that Hamlet has to face throughout the rest of the play.
The internal and external conflicts are closely intertwined in this play. It is mostly a play about the journey of a tortured soul to find peace with his duty to his murdered father and King through action.
Hamlet's main conflicts, both externally and internally, stem from the death of his father, King Hamlet, by his uncle Claudius. When, in Act I, the ghost of his father appears to his son, asking him to avenge his murder, Hamlet's life becomes rife with conflict and Hamlet's dilemma becomes the focal point of the entire play.
As a man of great thought, Hamlet's hesitates after his first reaction to the words of the ghost. For, he is reluctant to commit murder himself, knowing that "time is out of joint" and many other problems can ensue from his commission of a crime himself. And so, Hamlet begins his insistent existential debate within himself: To be, or not to be...." This debate develops into Hamlet's loss of will to involve himself in worldly affairs:
How weary, stale, flat, and unprofitable/Seem to me all the uses of this world!/Fie on't, ah, fie, 'tis an unweeded garden/That grows to see. Thing rank and gross in nature/Possess it merely (I,ii,133-136).
Because of Hamlet's ensuing melancholy, his inaction and disastisfaction with others aggrandizes and he perceives all others against him. He accuses Polonius as well as his daughter, whom he loves. of treachery; he trusts no one. His mother he describes as a villain:
O most pernicious woman!/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 (I,v,105-109).
In the end, of course, Hamlet does resolve his external conflict with Claudius (who has convinced Laertes to take his side) by accepting the challenge to duel Laertes. However, the greatest change in character is effected in Act V in Hamlet's assertion that he is "Hamlet the Dane" (V,i,227). Then, in the subsequent graveyard scene a new Hamlet emerges, a Hamlet ready for action no matter how it be directed by divine will or chance: "Why, I will fight with him upon this theme/Until my eyelids will no longer wag"(V,i,238-239).
A most complex character, Hamlet is all the more fascinating as he undergoes such a dramatic character change. Small wonder that "Hamlet" is Shakespeare's most popular play.