Updating from the road is hard. New Orleans is what New Orleans always is to tourists, there's lots of drinking and delicious food. But, onto the important matters at hand. I am going to make this short because this wireless internet keeps trying to crap out on me. While I admit that the setups are similar, the themes that run through the play and film are not and the film's director was not trying to simply retell Hamlet. For example:
The setups are very similar. Scar, brother to Mufasa (King of the African plains) kills Mufasa in order to become king himself. His nephew, Simba, runs away and then returns to fight his uncle for kingship. Meanwhile, Claudius, brother to Old Hamlet (King of Denmark), kills old Hamlet in order to become king. His Nephew, Hamlet (Jr.), avenges his father (eventually). I mean, it's got to be the same story, right?
Very different. Hamlet is plagued by the speed and unnaturalness of his mother who married his uncle after his father's death. He is then visited by a "ghost" who is supposed to be his father who tells Hamlet that Claudius murdered him in the garden. So, Hamlet spends the entire play agonizing over what to do based on what may or may not be his father's soul trapped in purgatory. His internal struggle is his battle with himself and his conscience, does he murder a man he does not know is guilty (there is a chance that this ghost is actually a demon in disguise trying to get Hamlet to commit a wicked act). Hamlet is never sure of Claudius' guilt except on the basis of circumstancial evidence (he never hears Claudius' confession while in prayer). Hamlet loses everything in his quest for revenge and everyone he ever cared about. His murder/death leave Denmark without a king and creates a power vacuum which Fortinbras (also avenging a father) readily steps into.
Simba instead is convinced he killed his father by calling out for Mufasa to rescue him when he was trapped by stampeding water buffalo. Mufasa would have escaped with is life but Scar secretly cast him back into the fray to be trampled by the water buffalo's hooves. His internal struggle is with his unreadiness to be king. He runs away knowing full well that Scar will rule for him. He only returns when he finds out what Scar has done to his homeland and his pride of lions (essentially handing it over to the hyenas). He returns because he has realized he must fulfill his destiny and become king. But he does not avenge his father, he grants Scar mercy, leaving him to "run away and never return" or Simba might then kill him.
In essence, the Hamlet story is about the many faces of revenge, blood in blood out, and the continuation of violence. The Lion King is about realizing one's destiny and not shirking responcibility. So while on the surface these two stories seem the same, the theme set them apart.