Accusations of improper relations with a goat.
Off the top of my head:
Elder brother is actually a girl.
Elder brother is actually a sentient speaking cactus.
Brother committed an unspeakable crime against the Kingdom/Empire or religion (if it has a lot of power). Prince comes up with 'undisputed proof'. There could even be a little bit of truth to it, though the prince could exaggerate it enough or have the King/brother framed for another crime - a real crime which was committed by the prince or someone the prince paid off.
Brother has a disease like tuberculosis which makes him weak/sick/bedridden all the time and gives the people doubts about his ability to rule.
Hmmm, not sure about this - the problem here is that most kings, at least as far as medieval-esque settings go, are absolute rulers by divine right, they're above the law (heck, they ARE the law).
Drakhov said:Hmmm, not sure about this - the problem here is that most kings, at least as far as medieval-esque settings go, are absolute rulers by divine right, they're above the law (heck, they ARE the law). He can make mustaches illegal (or compulsory), abolish Tuesdays and appoint his horse Chancellor - clearly as mad as a spoon, but still King. They can't be impeached, charged with a crime or voted out of office (i guess they can abdicate, but that's their choice, not somebody elses).
He CAN be accused of heresy, if such a concept exists in your society so that may be something.
Your prince, for whatever reason, seems to want his peoples support and approval as well as some justification beyond 'because I say so!' - here's an interesting idea for your consideration...
King is an absolute ruler, can't be accused of any crime.
Rules of Succession state the brother is next in line to the throne (assuming big brother doesn't have an heir)
Little brother walks up to big brother and outright kills him, then sits down to breakfast - he can do this in full view of everyone, since with big brother lying in a pool of his own blood, little brother is now King, no ifs or buts (despite the fact he's peeling an apple with the same blood stained dagger he drove into his brother's heart a few seconds ago, and is possibly giggling to himself or has his underpants on his head). After the initial shock wears off it's business as usual - "The King is dead! Long live the King!"
Fun fact: The current Swedish law says the monarch is not to have any political influence at all, to the point where he is not even allowed to express political opinions in public. However, he still enjoys complete immunity from prosecution.
That means that even thought our current king's role is entirely ceremonial, he can't be put on trial for any crime whatsoever. The cops have caught him speeding a few times, but all they can do is ask him to please not do that.
-Chuckle- I had the image of a man wearing a robe and a crown and sporting a big thick beard - in a sports car.