First of all, Welcome to the Chrons!
Does the prince -- the now-king -- know he killed his father and brother? I see you say he refuses to believe it, but on the other hand you refer to his guilty conscience leading to his mental breakdown which to me appears to be a little contradictory. Are you going to posit some kind of mental illness/dissociative personality disorder which prevents his truly knowing what he's done though he might suspect it? I have to say that unless you are a medical professional or have personal knowledge of such a condition I'm not myself too keen on that as an explanation, as it would be all to easy to trivialise mental illness. Actually, the whole "descent into madness" thing worries me somewhat unless you have a thorough working knowledge of mental disability.
I enjoy writing twist-in-the-tail stories, so I'd be happy to find out he is the real murderer, but in that case I'd want to see his search for those who killed his father and brother a Machiavellian ploy to try and convince others he is innocent. The problem with writing that, though, is you couldn't allow the novel to be written in his close POV, since he would know the truth of his actions. Instead, you'd have to write it all from other POVs, which would be a bit like Hamlet written from Horatio's viewpoint, possibly interesting, but perhaps in danger of falling a bit flat. Or, of course, you could write it from his POV showing from the start he killed them, and the interest then isn't whodunnit but can he get away with it!
As to the medieval-type world, it is perhaps overused as a fantasy trope, though there's always room for something well written! Are you a history buff? Is there a period of history which most interests you, and which you know back to front which could take the plot? I imagine it's a Western European setting you're considering, but in such a milieu would a king really disinherit a younger son? The elder would usually take the title under primogeniture, but even if in your society the father can choose his successor he doesn't need to disinherit the other -- why not allow the younger boy to remain Duke of Wherever, or marry him off to some distant princess to cement an alliance? Disinheriting a son is a big legal step for any nobleman to undertake, let alone a monarch -- it means throwing him out entirely without a penny -- so it's not something to be done lightly, and certainly not without a lot of consultation with the main Minister of the Crown/Chancellor/Grand Panjandrum/whatever the official who actually runs the Kingdom is called.
I have to confess to being dismayed when I read fantasies in which royalty and nobility act just like any regular guys. The rich are different from the rest of us, and the super-rich and powerful are very different. Apart from anything else, in most medieval societies they would never be alone -- either slaves or servants are constantly around, and even if dismissed from a room, they would be waiting outside, so true privacy is unheard of and consequently the opportunity to murder with impunity is difficult to achieve. So do think carefully when constructing your plot, and make sure your royals are really royal with all that entails! (Actually, I did wonder if perhaps it might make a more original story if instead of a king and princes you have a wealthy merchant and his sons. You'd lose the battle for the kingdom with the other noble family, but there's still scope for skulduggery and even warfare if you want to write it.)
That's probably enough of my thoughts to be getting on with! I'm sure lots of other people will pile in with suggestions, which will help you on your way. Writing is both an art and a craft, but you've come to the right place to learn both!
Good luck with the story.
