ok, and I actually already have a small plot used for the first season where the Fire Prince and his uncle had went on a trip together. so they are close and have shared some tough moments together. This brings some problems that weren't exactly there in Hamlet. In Hamlet, the uncle was a stranger that came into his life, taking his future career and life, but for Fire, the uncle is a true family to the prince. which makes this uncomfortable for when he confronts him about the truth. Also, the father had ben ill when the prince and the uncle left the kingdom, and the prince knows that his father is ill. which shows that the Prince is more worried about the responsibility of being king than the pain of loosing a father.
Also, the Fire Prince's name is Matthew