This is an interesting question. I ran a very long-running AD&D 2e/Call of Cthulhu mashup where we put a lot of work into realistic mental disorders from the PCs. As each PC went through more and more, they all started to display signs of mental disorders. (In fact, interestingly, it was a deep in character campaign and when I asked players to start playing their characters "a little crazy" it was remarkable that they were all different, they could all be pretty easily diagnosed using DSM-IV criteria, and the chosen disorder seemed to be the one that the player could tend towards IRL...)
First step is to check the definition of the disorder. From the DSM-V, schizophrenia consists of:
Two (or more) of the following, each present for a significant portion
of time during a 1-month period (or less if successfully treated). At
least one of these should include 1–3.
- Disorganized speech
- Grossly disorganized or catatonic behavior
- Negative symptoms (i.e., diminished emotional expression or avolition)
It can also help to look at well realized examples of schizophrenia in other media - the movie Clean, Shaven is my favorite here.
Now, of course trying to depict someone very mentally ill can be a challenge to also be functional - what we did was to start with the schizotypal personality disorder. This is a more mild form and often a 'stepping stone' to full blown schizophrenia. It's a little more accessible to the layman as well, here's the list of possible symptoms:
- Inappropriate or constricted affect (the individual appears cold and aloof);
- Behavior or appearance that is odd, eccentric or peculiar;
- Poor rapport with others and a tendency to withdraw socially;
- Odd beliefs or magical thinking, influencing behavior and inconsistent with subcultural norms;
- Suspiciousness or paranoid ideas;
- Obsessive ruminations without inner resistance, often with dysmorphophobic, sexual or aggressive contents;
- Unusual perceptual experiences including somatosensory (bodily) or other illusions, depersonalization or derealization;
- Vague, circumstantial, metaphorical, over-elaborate or stereotyped thinking, manifested by odd speech or in other ways, without gross incoherence;
- Occasional transient quasi-psychotic episodes with intense illusions, auditory or other hallucinations and delusion-like ideas, usually occurring without external provocation.
In our case, the player of the schizophrenia-tending wizard started to behave and appear "oddly" - dressing inappropriately (women's underwear under his robe), having trouble forming relationships (paid another PC to pretend she was sleeping with him), collecting eyeballs (somewhat plausible as a wizard hobby, but...), belief in a personal deity that wasn't externally recognized (lived in fire), extreme suspicion of the blind gnome in the party. He'd also follow the orders of one party member without question. Whenever he had significant setbacks he'd have audial hallucinations that would "drown out" what others were saying and could make him exclaim things like "SHUT UP SHUT UP!" at times, which was very disconcerting to his fellow party members. He did a really good job of it, didn't lay it on too thick in normal circumstances unless he was failing a lot of Sanity checks, but it was clear and reasonably realistic. (The other party members were a megalomaniac, borderline personality disorder/bipolar, a depressive/catatonic type, and an anxiety disorder).
So in general I'd recommend starting from a deep in character perspective - mental disorders become goofy stereotypes if the character is a goofy stereotype already. Research the disorder, including its milder forms, and slide into it.