After a few weeks in Milan Prospero starts to remember how useful magic was and how great that power felt
Prospero enters
After his sudden realization Prospero travels to the kings castle by horse because it was the fastest way for him to get there
king dies
Prospero arrives at the castle, and is let in because Alonso now sees him as a friend after what happened on the island, and because he doesn't pose a threat because to Alonsos knowledge he gave up magic
Prospero takes his place as king
When Prospero is let into the castle he sends a giant ball of magic at the king from his new magic staff, which is gold from the money he has access to as a Duke.
Prospero has successfully slain the king, and with Miranda and Ferdinand out on honeymoon from the marriage Ferdinand is not here to take his place as heir to the thrown so Prospero takes over
Prospero now having completed his revenge plot, takes his spot as king. This was his end goal all along, nobody every truly can change