People with Psychosis sometimes get "Delusions of Grandeur", which can lead them to believe they are gods, or the next Jesus. People suffering from the 'mania' part of bipolar disorder often describe these thoughts as being perfectly logical at the time.
Then there is the Jerusalem Syndrome, where normal looking people would go to the city of Jerusalem and declare that they are Jesus or one of the biblical prophets. The Kfar Shaul Mental Health Center of Jerusalem treats 100 of these patients per year.
In a highly "amusing" case of medical un-ethics, psychologist Milton Rokeach once placed three schizophrenics who all believed they were Jesus into a room and let them talk to each other. After some initial violent arguments on who is holier, all three men individually decided that he is the Christ, the other two are schizophrenics in a mental hospital. Rokeach published the book The Three Christs of Ypsilanti about this case.
There was the Greek philosopher Empedocles, who, without a doubt, thought he was god. An active volcano disagreed. Strongly.
There's also the legend that Empedocles knew he wasn't really a god, but wanted to fool people into thinking he was one by claiming to ascend to Olympus but actually jumping in the volcano (so his body would not be found). Supposedly, the volcano thwarted his scheme by sending one of his sandals back up in a minor eruption, so people knew he jumped in.
Basketball player Josh Childress said that during his experience for playing with Greece (more specifically, the Olympiakos), he ended up being treated as a god towards the country. When Josh decided to leave Greece to return to the Americas (as a member for the Phoenix Suns), some citizens got a little bit hot-blooded.
Pastor Apollo C. Quiboloy. To quote: "Pastor Apollo C. Quiboloy is the Appointed Son of God in this last days." Since the Son of God (Jesus Christ) is God the Son, this cult leader essentially considers himself God.
During their bout at Madison Square Garden on March 8, 1971, Muhammad Ali repeatedly taunted Joe Frazier by asking "Don't you know I'm God?" Frazier replied, "Well, God, you gonna get whupped tonight!" and then knocked Ali out.
Some people with mental illness may develop and believe in delusions that they are a biblical figure or a god/deity reincarnated. A few examples include:
Jim Jones, founder of The Peoples' Temple which essentially became a cult with him as the ringleader. He claimed to be the reincarnation of Jesus, Buddha, Akhenatan, Vladimir Lenin, and Father Divine. He organized a mass suicide at Jonestown in Guyana by forcing his followers to drink Flavor-Aid laced with cyanide; it claimed the lives of 918 people.
David Koresh, leader of a Branch Davidians sect believed to be the final prophet AKA Jesus himself. A raid on his ranch in Waco, Texas in 1993 eventually started a fire. The aftermath left 52 adults and 21 children dead, including Koresh himself.
The above 3 could all just be cases of narcissistic and / or anti-social personality disorders, ie. The Sociopath, meaning that they were all just Manipulative Bastards who told people that for money, power and For the Evulz. Jim Jones for instance used to be a con-man who reportedly once told a fellow criminal of a scheme to swindle people with a new religion, while Charles Manson has also claimed he was simply "nobody", and isn't considered insane by the state of California since he was simply locked up in prison for the rest of his life.
Most sociopaths and all narcissists have this as the core of their psyche.
As mentioned before, some people with mental illnesses such as schizophrenia become convinced that they are the Second Christ, Buddha, Mohammed, or God himself.
Christopher Roller sued David Copperfield and David Blaine in 2005 on the rather ludicrous grounds that they were infringing on God's powers, and that Roller himself is God. Who does he think he is, a Holy Roller?
Inverted by Dr. Robert Oppenheimer, commonly acknowledged as the inventor of the Atomic Bomb. Rather than celebrate the accomplishment, he anguished over its destructive power (having witnessed it firsthand), publicly and famously lamenting, "Now I am become Death, the destroyer of worlds," quoting Hindu scripture.
A great many dictators have this as their pathology such as Adolf Hitler who wrote hymns proclaiming himself as the saviour of Germany, ordered that a copy of Mein Kampf be placed on every altar and planned to have himself declared the new Messiah after conquering Rome.
In an inversion, it is often said that Augustus Caesar specifically had a slave ordered to follow him and whisper the words "memento homo" (roughly translated as "remember you are human") to him in order to ensure that the praise and fawning did not go to his head. Didn't stop him from minting coins identifying him as Divini Filius, the son of (a) god.
This is an older tradition. According to Tertullian, a Roman imperator (general) on his triumph would ride before his troops in a chariot, dressed as Jupiter. Behind him would stand a slave, holding a laurel wreath over his head, and whispering: "Respice post te! Hominem te esse memento! Memento mori!", meaning "Look behind you! Remember you are human! Remember death (that you are mortal)!" This was done to avoid blasphemy.
Jett Simmons McBride thought he was Jesus and ran someone over with a car. He was stopped by a hitchhiker with an axe.
Matt Thredening has formed a cult for worshipping the supposedly lovecraftian-esque Will Ross-Girard.