Psychosis of any kind is indicated by an episode where an individual loses contact with reality; it could be in the form of a hallucination, or any other form of lost consciousness caused by internal mental processes gone wrong. It could range from a chemical reaction such as drug and alcohol poisoning, to genetic issues, as well as the onset of several possible mental conditions.
This being said, a psychotic episode can happen to people who are NOT mentally ill. As it was just stated, psychosis can be influenced and caused by a number of variables ranging from internal to external stimuli. Concisely, psychosis is not a diagnosis, but a symptom or event that occurs to someone who temporarily loses contact with their immediate reality.
In contrast with psychosis, schizophrenia IS a diagnosis. It is a mental illness that in many cases (20% of them) is inherited. However, to meet the criteria for schizophrenia you would have had more than one psychotic episode, plus manifest another form of delusion within a small period of time, often at three months.
According to recent studies, the onset of schizophrenia and psychotic episodes both tend to occur during adolescence given the unique and delicate state of development of the teenager's brain; that, combined with hormonal imbalance, social pressure, and any inherited traits can combine to form chaos. Yet, studies also show that many of these delusional and psychotic episodes are not necessarily pre-schizophrenic, nor do they manifest in the same matter as schizophrenia. Hence, many times adolescents fall under a NOS diagnosis (Not Otherwise Specified) of psychosis much more often than a diagnosis of schizo. As a result, the latter has replaced the later in the psychiatric community as a much more common trend among adolescents.
In conclusion, it may be that we, as non-experts, tend to use psychotic and schizophrenic as to interchangeable adjectives. While a schizophrenic person will undoubtedly experience episodes of psychosis, a psychotic person may just be under extreme stress, grief, PTSD, a drug trip, suffering from extreme lack of sleep, undernourished, overworked, abused, or "temporarily unstable". It is more likely to have a psychotic episode than to be diagnosed with full-blown schizophrenia. The complexity of this disease is so great that it needs to be diagnosed responsibly and accurately so that the proper interventions can take place.
How a person can self-diagnose Schizophrenia, i think it is better to consult a psychiatrist.