I'm using "psychotic" here under the definition of 'a break from reality,' usually accompanied by symptoms like hallucinations, delusions, distorted perceptions or reactions to perfectly normal stimuli, and so on. That means that this is not advice geared towards writing a character with a specific psychotic condition like schizophrenia. For one thing, I don't know enough about them. For another, in a fantasy world, a writer might have a condition that doesn't match any Earthly mental illness, so it wouldn't do much good, anyway. It's always better to do research on a specific disorder if you want to use it, and not rely on general advice.
1) Be as immersive as possible. It's probably going to be difficult to explain what I mean by this...no, wait, I think I've got it. An immerse viewpoint experience convinces readers that this is what it's like inside the character's head, of course, but it also immerses the author. Whether she really believes in that character's perceptions or not, she can believe them while she's writing them.
Does all writing need to be equally immersive? No. I've read omniscient stories and stories that didn't use a very tight third-person, and still found myself utterly convinced of the characters' reality. However, I've also read many half-assed stories where it was obvious what the author believed, especially when two characters argued or engaged in philosophical debate. One character's argument was presented as almost self-evident, while the other character argued with stupid themes that even he or she acknowledged were stupid, and had half-surrendered before the debate began.
This, I think, is the author not being immersive enough. She's not able to let go of her own philosophical, religious, or moral ideals enough to think and feel and believe as someone else while she's inside that person's head. I don't mind if she does write one character more strongly, as long as that character is strong for some reason other than just because the author agrees with him. And I don't mind stories that treat everyone with the same consideration (or lack of it, as in a parody). I do dislike it when the author doesn't have an equal commitment to everyone involved and tips her hand so that I can tell it, by the way she writes.
Now imagine doing a half-assed immersion with a psychotic viewpoint character, whose reality is not the same as every other character's reality and has to be shown that way.
I do think an author has to have a commitment to this, or she's simply not going to achieve anything worth mentioning, and I'd be kind of puzzled why she wanted to try. And that means letting go her own restraints and falling, or diving, or ascending, or using whatever metaphor you want, into that perception of reality.
If you want to write a psychotic viewpoint character, see if you can believe what he believes before you enter his head, even if only for that time. It would help.
2) Decide how to reflect his voice via narration and dialogue. This will probably depend on what person you're writing. However, if you're writing omniscient, then you're probably creating a tone and style of narration that's separate from what any one person in the story thinks or feels anyway, so I'll talk instead about third and first-person.
Why would a psychotic character have a disjointed inner narrative but dialogue that was perfectly normal and responded in perfectly normal ways to what went on outside his own head? Maybe you can come up with a reason, but if so, decide on it, and decide how you're going to show it. I might take a character who was perfectly with it on the surface and not with it inside his own head as surreal, but not as psychotic. After all, he's obviously still able to function and convince people he's normal, and that suggests that, on some level, he's aware that his beliefs could be false or delusional.
Some methods I've seen work at conveying psychosis:
-Fixation on some piece of trivia or nonsense or lore that means nothing to anyone else (a nursery rhyme, the fact that dragons' eggs are blue).
-Absolutely powerful convictions that never come true (here, the author needs to be skilled in showing that the character could be wrong).
-Tricks with memory (perhaps the character's beliefs never come true, but he never notices, because he rewrites the past in his own head all the time).
-Disassociation, though in this case it would need to be permanent disassociation.
-Extreme reactions to innocuous dialogue and/or situations.
-Contradictions piled on each other.
Once again, there's a challenge here. The author is trying to convey two realities, the one inside the character's head and the one outside, and she's trying to do it while making the one inside the character's head appear as 'real' and immediate as possible- while it's not true. It's easy to write a story in which the character just appears to be nonsensical, or in possession of some deep wisdom that no one else realizes (that one happens all the time with supposedly mad characters in fantasy). It's much harder to write a true psychotic from inside her own head.
3) Strive for description. Dealing with hallucinations? There's an opportunity for vividness- not necessarily of the dancing pink elephants school, because the visions might be dull in color or affecting another sense than sight, but they won't be the same as what other people see. Also, unlike writing a character who does mostly deal with consensus reality, you can't count on your reader "knowing" what the psychotic character experiences without much description, as you can when you describe someone picking up a telephone or pausing under an oak. Description is a must to orient your reader in the character's world.
Dealing with delusions? These are overwhelming, not just the mild conviction that someone is watching you or that your neighbor cheats on his taxes. Go in-depth. Work out the chains of crooked logic behind them. Know what and why the character believes at least as well as you'd know the true philosophical/religious bent of any other protagonist. Try tracing back the origin, and seeing why the character might have come to have delusions focused on this one particular thing.
Dealing with paranoia? Find it everywhere. Relate the most innocent objects and dialogue and actions back to it. Show the character's typical response to it- terror, fury, despair, world-weary resignation, or something else- in as much detail as you can.
Description of psychotic phenomena can tax an author's skills, but, if done correctly, it's probably the single part of writing most useful to showing your readers that this character's reality is not theirs.
4) Make it a sealed mental world, or at least as much like one as possible. If the character is only psychotic some of the time, then you can skip this point, but I'd be puzzled by a character who seemed to doubt that his hallucinations and delusions were true, or was even capable of rationally considering the evidence against them, and yet continued to believe them.
What escapes would someone usually use to get out of a frightening or highly emotional or dream-like state? Testing the physical state of objects might be one, pinching themselves or stamping on the floor to see if it's solid. Of course, a psychotic character probably won't be receiving reliable input from his senses, either, and if he's spent long enough in this state, he may have ceased to trust them, or just accepted what they tell him as normal.
Another way out might be dependence on another person. But can anyone actually reach the psychotic person? Here's another plot in which mental healers could be interesting, if only they didn't succeed without effort. The psychotic character is not likely to have a lot of trust in someone else even if they really can be trusted. And, of course, if a mental healer enters a psychotic's inner landscape, she's not going to have any form of orientation in it at all.
Or perhaps this world really does seem solid and seem to make sense to the psychotic character. I'd argue for that one, because it's the most challenging to write, and fun if you can make it work. In that case, he has no urge to try and "wake up," and the entire story is filtered through his unique perspective. Only fitting, really, since it's his viewpoint.
5) Changeability/mutability. Someone who's psychotic may have emotions that change very quickly, especially in response to stimuli from the world outside his brain. Try writing them as normal, rather than having the character pause and introspect (as is normal for fantasy narrators) to try to work out why he's angry or angsty or in pain. It'd go a long way towards setting up the feel of someone who can't think in any remotely real way about what he's experiencing.
Perhaps his emotions are steady most of the time, like a paranoiac's terror, but his actions are wild. He tries stabbing one "enemy," shooting another with an arrow, jumping on another person's head the next. Or a shadow-monster is fought one way, and the demon that his hand turns into another. Here, he might introspect and rationalize his actions, but once again he's not going to spend a lot of time pausing and thinking about why they're strange according to the standards of other people.
What does it do to a character to constantly change, and either not notice or find rationalizations for it? Another possibly good thing to explore with a psychotic viewpoint character: writing someone who's not very self-aware. Once again, damn few fantasy narrators fit that category, as they're often explicating their surroundings as an introduction to the fantasy world for the reader, or explicating their emotions so that the reader doesn't miss that someone's supposed to be falling in love here. Writing without that quality is very challenging, but then, writing a psychotic character is a challenge anyway.