I found a video with all the comparisons, because there are too many parallels to count.
So here's the main breakdown:
Simba - Hamlet
Mufasa - Old Hamlet
Scar - Claudius
Sarabi - Gertrude
Nala - Ophelia
Timone & Pumba - Rosencrantz and Guildenstern
However, I would like to point a few contrasts. First off, Timone and Pumba are not killed by Simba. Nor, as we all know, does Nala drown or Sarabi die. I think the most obvious one of all is that Simba does not actually die, and the circle of life *cue music* is completed (rather than in Hamlet, where everyone basically dies).