The plot of The Lion King is based on Hamlet, and so is Lion King 2 based on Romeo and Juliet ;)
Actually, although the plot of The Lion King is said to be based on Hamlet, it doesn't have very much in common with it. We have a king who is murdered by his brother, who usurps the throne, and is avenged by his son. This is a very common type of plot. Shakespeare used it more than once. In Macbeth we have a king who is murdered by his cousin, who usurps the throne, and is avenged by his son. In Richard III we have a king who is murdered by his brother, who usurps the throne, and is avenged by a distant relative. The pathetic fallacy that the kingdom actually decays under the rule of the usurper is present in all three of these plays (and the Lion King).
What else about the Lion King may be similar to Hamlet? There is a foolish counsellor, who does not get killed by the Prince. The Prince has a girlfriend, who does not suffer grief for the death of her father, become insane or die. She doesn't have a brother either. There is a sort of ghost thing in the sky.
In Hamlet, one of the characteristics of the story in all its incarnations is that the Prince is forced to remain in the court of his usurping uncle. He is forced to adopt a pose of craziness to make himself appear harmless. There is none of this in The Lion King. In fact the idea that the Prince flees believing himself to be a suspect in his father's murder is right out of Macbeth.
In Hamlet the Prince dies in accomplishing his revenge. Not in The Lion King. Again, it's more like Macbeth.
In Hamlet the Prince has one special pal who is stable, logical and cool. In The Lion King he has two friends who are Abbott and Costello. The Prince's mother marries the uncle willingly in Hamlet; in The Lion King she is enslaved. In Hamlet the Prince only accomplishes his revenge at the last minute and without planning; in The Lion King he and his sidekicks have a sneaky plan to get into the castle (sort of like disguising themselves as Birnam Wood) to effect the revenge.
In fact the more you think about it the less like Hamlet the Lion King seems. It's at least as similar to Macbeth, and probably more.