Short answer: yes, though those we know about were kings and powerful men, not insane people. This was a particularly common practice in the Hellenistic period. Reactions could vary, but generally were positive, at least when the person was still alive. Indeed, often it was the case that a man did not claim divinity, but had divinity thrust upon him by a group of grateful people. The genuineness of believe in such gods is impossible to gauge. Belief is internal and its outward manifestations can be interpreted in a variety of ways: is dedicating a temple to a king an act of genuine piety or an act designed to cynically curry favor? Our own modern view of what constitutes "authentic" religious devotion is heavily influenced by Christianity and its idea of a personal and private relationship with a god. We need to be aware that there is a difference both in form and in terminology between our religions and the religions of the Greeks.

Long answer: The first of the Greeks of whom we know to receive divine honors while alive was the fifth century Spartan general Lysander. So pleased were the men of Samos that he had broken the power of the Athenians that they offered sacrifice to him as to a god. Now, he was not the first man to have received sacrifice, for the heroes of old received sacrifices too, and the men who founded colonies were worshiped as heroes in the cities they founded after they diesd. But heroic worship and divine worship differed slightly in form and terminology, so Lysander is different in that he was both a recipient of divine honors and a living recipient of sacrifices.

There are a scattering of fourth century examples of divinization. The most famous is of course Alexander the Great, who apparently came to think of himself as divine after a trip to the Siwa Oasis in Egypt. After Alexander, the pace of apotheosis quickened. First came the kings. Ptolemy I was proclaimed Savior (Soter) by the Rhodians after he helped deliver their city from siege. When Demetrius Poliorcetes "freed" Athens from a rival king, the Athenians offered him an array of divine honors, including the right to live in the Parthenon. The following is from a famous hymn sung in Demetrius' honor:

"For other gods do at a distance keep, Or have no ears, Or no existence; and they heed not us - But you are present, Not made of wood or stone, a genuine god. We pray to you."

As time went on, the idea of a ruler cult ossified in the Greek cities. There were two basic branches to the cult. The first was composed of the piecemeal cults created voluntarily by cities for individual monarchs, usually as a sign of gratitude for some great favor. This was augmented by a second formal cult promulgated by the kings which included sacrifices and holy days for all the past members of the royal house, and the most important of the living ones. This cult was certainly followed at court, but it can be hard to tell the degree to which it was followed by subject cities (we know most of this from inscriptions, so our knowledge is both fragmented and limited).

In the later Hellenistic period, other men began to receive lesser (heroic) honors. The second century musician Kraton, was given a statue in the theater of Teos, to which incense was offered on the festival of Dionysus and on certain festivals of the royal cult. After Kraton's death, the members of a musicians guild he had founded and endowed paid him heroic honors.

The general idea here is that if a man does something so absolutely incredible that he can never possibly be repaid in kind (by, say, destroying the Athenian Empire, or the Persian Empire, or even just being incredibly generous) then the appropriate response was to honor him as if he were a god, since gods too bestowed such unrepayable favors on men.

There could be a backlash against divine honors. For example, the Greek cities wished to give Augustus divine honors (because that was what you did for kings), but Augustus was extremely leery of this sort of thing, for Julius Caesar had dallied with the trappings of kingship and ended up dead because of it. Eventually a solution was reached: Augustus could be worshiped in a few carefully chosen temples in the provinces while living. After death he was worshiped as a god everywhere, including Rome. The benefit of hindsight also made some divine honors seem excessive. Plutarch, for example, inveighs heavily against the Athenians for the honors they gave Demetrius. He thought they went overboard in a desire to flatter their king.

Good reading (since you are interested in Greek religion is W. Burkert's classic book titled, appropriately enough Greek Religion. If you want to know more about Hellenistic ruler cult, the article in the Cambridge Ancient History (1984) gives a good and concise overview of the topic. For the Roman emperor's cult in the East, S.R.F. Price's Rituals and Power is good.

I'll also speak to the Roman case. Others have mentioned examples like Commodus claiming to be Hercules and the veneration of emperors as gods, using the title divus. However, during the principate it was very rare for an emperor to claim to be a god. It was much more common to claim to be related to a god. This started with the deification of Julius Caesar and Augustus's claims to being "divus filius" or son of God. This continued as many emperors were deified after death (Vespasians final words were "I think I am becoming a god") as it was useful for emperors to claim divine descent to solidify their rule. This is similar for the case of Commodus who claims to be Hercules, not a god himself but the son of a god.

The main counter example during the early empire is Domitian who claimed himself as "dominus et deus" or lord and god. This is indicative of his more authoritarian rule and why the aristocratic classes didn't like him even though he was a good administrator and popular amongst the lower classes.

During the Dominate, the latter portion of the empire, the emperors would adopt Domitian's titles, notably the dominus one as dominate is derived from it. Deus fell out of favor as the empire became Christian following the end of the tetrachies and rise of Constantine's dynasty.

In all, it was not very common for Romans to claim to be divine in life, though emperors were often deified after death.