In the play Hamlet what are Hamlet's external conflict and internal conflict?

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.