Hamlet's quest is to kill his uncle, as evidenced by his trials and soliloquoys throughout the play.
Hamlet's struggles are either over whether he is killing the right person, if he should kill his uncle, or if he should give up and kill himself there.
Unlike in The Lion King, Hamlet is told by his father his quest from the very beginning, even if he doesn't believe it completely. While he also has to get over his indecision, this would not be considered his true "Quest", merely a part of it.
Quest in Hamlet vs Lion King
On the other hand, Simba's quest is to find himself.
This is evidenced in the contrast between his opening and mid-movie selves. He begins with the song "I just can't wait to be king", and yet, as a young lion seems to have lost interest, or be afraid.
Seeking to repair his lost and broken self, he takes and easy way out and adopts "Hakuna Matada". While allowing him to get over his loss, this motto dooms his sense of responsibility and hinders his discovery of his true self.
It is not until Rafiki "enlightens" him with the help of his father's ghost that he can discover his true self and fight scar.
The reason killing scar is not his quest is obvious; there is no focus on his confrontation, and he does not realize scar is his enemy until near the end of the play.
