I am assuming you are referring to government forms when you say the 'nature of the state'. In this case, the answers are often very different depending upon the polis in question, but we can make some generalizations. Let's just assume there are four main government types to talk about: tyrannies, democracies, and oligarchies, and monarchies. Sparta is a bit different, but I'll talk about that too. The first three arose in a vaccum from the lack of the last one.
In general, most of the cities were in a state of flux as different forces within the polis battled one another politically (and sometimes militarily) for supremacy of their government type. Each of these government types had general types of backers:
Based on this, we can start to see how these conflicts played out in general. If a polis had a large number of free land holders, democracy was the most powerful force. If they had a smaller number of aristocratic land-holders who dominated the rest of society, then oligarchy was the the likely result. And finally, if a single man/family came to dominate, a tyranny was the likely result. If a tyrant could maintain power long enough they might claim to be king; it is important to remember that tyrant didn't necessarily claim any negative connotations in ancient Greece as it just meant someone who came to executive power through unconventional means. That said, tyrants often came to power not simply internally but externally. There are numerous tyrants in history who were placed there by external actors, and as one might suspect, if that external support disappeared their hold on power often did as well.
There is also one peculiar example, and that is of Sparta. Sparta as it is known in popular culture did not always exist as it did. It became the highly militarized culture because of specific circumstances related to it's expansion; as it came to dominate the Peloponnese, the balance of manpower was such that farming and landholding, the traditional powerbases of the aristocrats in other poleis, was carried out by serfs/slaves (the Helots) while Spartan citizens were fully mobilized at all times for war. This means that the power dynamic in other poleis, that is, between the free landholders and aristocrats (and the aristocrats with each other) didn't really exist.
So, summing up, we can make a basic generalization of the cycle of changes in a Greek polis. A state might be a tyranny, then that tyranny might be overthrown and either an oligarchy or democracy imposed depending upon which class (the aristocrats or free holders respectively) could muster more power and support. Then it might shift between them as power wained. If a particular man/family became powerful enough, either through domestic or foreign support, he might seize power personally. If he could retain that power, his family might become royal (as in the case of Syracuse). If that family is overthrown, we start the cycle over again. And around we go.
/u/Spoonfeedme gives an answer, but I think I will try to give a different perspective. I am assuming you are taking about the so-called Archaic Period, which was essentially from the eight to fifth centuries BCE. Traditionally, this was viewed as the period in which Greece emerged from the shadows of the post-Mycenaean "Dark Ages", and although that narrative is of course deeply problematic there is some value to thinking of it as the crucible that forged the contours of later Greek civilization. This period saw the early developments of Greek literature, colonization, the full formation of the polis and the emergence of the civic political structures. These in turn are bound up in the social and demographic trends of the time, particularly the increase in population and the corresponding increasing social hierarchies, which refers to both the gap between rich and poor and the concentration and centralization of political power, often into particular elite family lines. This lead to the increase of scope of civic and state formation and of social conflict, which I will put into the Greek concepts of stasis and synoikismos.
Synoikismos is more or less "within the same household" and describes the process by which the scattered communities in the countryside conglomerated into the polis. The polis is a city, of course, but also has very strong elements of social grouping, shared culture and identity, so it goes far beyond a mere political or demographic structure. The classic example of this is Athens in which, traditionally, the hero Theseus united all the outside communities of Athens into a single structure by killing monsters and bandits and establishing the cult of Athena Parthenos on the Acropolis. Now, naturally, it is unlikely that this is a literal description of what actually happened, and in fact the entire myth cycle of Theseus is kind of artificial, but it gives a good idea of how this was thought of by the Greeks themselves. The modern perspective relying on archaeology sees an increase in the population of the countryside and the corresponding increase in individual wealth and power. This in turn lead to an increase in the demographic concentration into the new cities (although many of these were actually the site of old Mycenean cities). The frictionless, perfectly spherical model at STP and without wind resistance sees the polis (city) developing out of the oikos (household).
This process, however, was not an easy one, and going along with the increase in wealth was an increase in poverty, as poorer people fall into debt bondage or urban destitution. Furthermore, the increased political centralization also saw an increased political competition, as elites vied for the ever increasing prize. This also produced winners and losers, and these two processes in tandem lead to stasis, the conflict between different elite factions and between different social groups. This is what lead to the emergence of the new political types, such as what /u/spoonfeedme described, although in general it is often imagined as changes from an initial structure of oligarchy or aristocracy. The classic narrative is the emergence of the tyrants, who were generally speaking members of aristocratic clans who were able to rise above their rivals and assume absolute power, very often through the support of the people, although this was by no means the only path.
It is worth noting that there was no correct or single way for a Greek region to proceed, and the models tend to be irreparably biased by the example of Athens. But in Boetia, for example, while Thebes was able to emerge as the major political center it was never truly able to effect synoikismos and bring its rivals to heel. Thessaly never even developed a polis, instead creating what was termed an ethnos, or essentially semi-urban confederations, and lest we think this si somehow less developed, remember that the Thessalian tagus Jason was considered the great hope of pan-Hellenists seeking Greek reunification.
The classic study is Anthony Snodgrass' Archaic Greece, and like most classic studies it has been chewed up pretty badly in the past decades, largely for downplaying the achievements and importance of previous periods, but has still received some support by archaeologists such as Ian Morris. For a good, introductory overview of the topic, as always consult Sarah Pomeroy's Ancient Greece: A Political, Social, and Cultural History. I disagree with just about everything in it, but it is an excellent starting point.