Of course I have to write another Lion King post! The popularly accepted belief among internet users and Lion King fans is that the movie The Lion King is most similarly related to Hamlet the play. The most glaring similarity, I believe, is the relationship between the prince and the evil uncle. Both Scar and Claudius (the brothers of Mufasa and Hamlet Sr., respectively) were jealous of their brothers' power over the kingdom and killed them in order to acquire that power for themselves. Simba and Hamlet, the princes, are, therefore, both fatherless :( and Mufasa and Hamlet Sr. each appear to their sons in the form of ghosts. Both Simba and Hamlet have love interests--Simba develops a crush on Nala, while Hamlet develops a relationship with Ophelia.
Since we haven't read the whole play yet, I cannot write every similarity and difference there might be, but I will probably comment on this as we read further with updates!