This question might be different based on different city-states, but I'm interested in any pertinent information.
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Great question. There was indeed a rank structure, which did indeed see a lot of variation, so to keep things clear I'll break this down into the best attested categories: Athenian militia, Spartan militia, and mercenaries.
The main thing to bear in mind is that Greek states generally did not have standing armies. Their forces were mustered when the need arose, levied from the citizen population. Because of the needs of everyday life (the agricultural calendar in particular), most of these armies could not exist for very long before its warriors had to return home. Large levies could serve for perhaps 60 days at most, and usually much shorter than that.
This is relevant for two reasons. First, it means there was rarely much opportunity to "rise through the ranks." Armies usually weren't on campaign long enough to allow time for such a process. Second, commanders couldn't afford to wait until talented candidates for higher ranks presented themselves. They needed the basic cadre of subordinates ahead of time, because it was the officers that would muster the required number of men from the various districts of their community. The result is a system that looks like the inverse of that used in modern armies: the officers were picked before the troops were enlisted. Armies tended to be mustered by their officers, filling the ranks of their units as they saw fit. After that, any rearrangement of the chain of command was more or less unthinkable; no officer would opt to replace himself.
In other words, there wasn't really such a thing as "promotion" in Greek armies. But that doesn't mean that it would be impossible to gain a higher rank due to merit. First (as we'll see below), some armies did serve for longer periods of time, and gaps might open up in the hierarchy for promising candidates to fill. Second, citizen armies may have only existed briefly with a predetermined set of officers, but they were raised annually in wartime. Men who had proved capable in one campaign might see themselves preselected for a higher rank in the next. That is the kind of "promotion" that ambitious Greek men of war would typically seek.
With that general context in mind, let's get into it.
Athens
Ignoring the Archaic office of the polemarch ("war-leader"), which became ceremonial in the Classical period, the Athenian hoplite militia had the following rank structure (with equivalent ranks existing for the archers and cavalry):
Among the many civic offices of the Athenian democracy, military commands were unique in that they were filled by election, and there was no limit to the number of times an individual could hold them. Both features were generally seen as deeply undemocratic. Military offices were the exception to the rule of filling offices by lot for short terms because it was recognised that skill and experience were important; not everyone was equally suited to command.
The board of 10 generals, 2 cavalry commanders, and 10 tribal commanders were elected each year by the Assembly ([Arist.] Ath.Pol. 61.3). The generals could come from any of the 10 ‘tribes’ (administrative districts) of Athens, but the tribal commanders were each a representative of their own tribe. Their job was to muster its taxis, its share of the levy (which could include well over 1,000 hoplites) using the deme registries or hand-picking men from the muster rolls. To help them out, the tribal commanders would select their own lochagoi, whose number is unknown. Probably there were only 2-3 per taxis, each commanding several hundred men. All of these were selected before a single citizen picked up a shield.
As you can see, there are several ways to gain rank in this system. If you distinguished yourself in earlier campaigns, or if you just happen to be good buddies with your tribal commander, you might get yourself picked for the lowest rank of lochagos. That would only require convincing one person that you were good for the job. On the other hand, if your record convinced many people of your abilities, you might find yourself voted a taxiarchos or even a strategos. Any citizen over 30 was eligible for these offices. The choice was often not based on very substantial reasons: the tragedian Sophokles was supposedly elected to the generalship once because he wrote such good tragedies about war.
One peculiarity of this system is that you don’t really need to climb the ladder; you could be elected straight to a generalship if you could convince enough people that you’d do well as a general. But it usually helped to have shown yourself useful in a crisis. Previous tenure as a lochagos or taxiarchos would probably help, and young rich men deliberately tried to get into the public eye through the cavalry command in order to build their reputation for an eventual generalship. Ancestry was less important in theory, but in practice Athenians still paid a lot of deference to the old leisure-class clans, so it certainly wouldn’t hurt to have an Alkmaionid or something in your bloodline somewhere.
In short, the answer for Athens (and, as far as we can tell, other democratic Greek states) was that there was indeed a rank structure, though it was very limited; you could indeed rise through the ranks, though it was a long game and usually required additional advantages such as pedigree and oratory skill. Those were the rules of Greek democracy. Only truly exceptional individuals like the low-born Iphikrates could rise from marine to peltast commander to preeminent strategos on merit alone.
Sparta
The Spartan militia had a similar structure to the Athenian one, but it was complicated at both ends by unique features of Spartan society. At the top, the highest commanders were always the hereditary kings. At the bottom, the translation of Spartan tent groups into military units and the need to integrate non-Spartiate warriors into the phalanx (due to constantly dwindling numbers of actual Spartan citizens) produced a more detailed unit subdivision. The result was a hierarchy of officers that astonished other Greeks with its detail and complexity. It looked roughly like this:
Thucydides described the resulting phalanx as a formation of “officers over officers,” in which nearly the entire front rank consisted of commanders and sub-commanders. Undoubtedly there were many more officers in a Spartan army than an Athenian one. Xenophon (Lak.Pol. 11.4) explains how the Spartan militia of his day was divided into 6 morai, meaning a total officer hierarchy for hoplites of 2 kings, 6 polemarchs, 12 lochagoi, 48 pentekonteres and 96 enomotarchoi. In this hierarchy the polemarchs have a similar function to the Athenian tribal commanders, and they were probably similarly responsible for mustering their mora. But if there were no kings in command of a particular Spartan force, the polemarchs could also be generals.
Unfortunately we don’t really know how these ranks were filled. On the one hand, competition for status was one of the most fundamental tenets of Spartan society, and there can be no doubt that young Spartiates would have competed fiercely for every rank in the hierarchy. On the other hand, prosopographical evidence (analysis of names and family ties) shows that most of the polemarchs were extremely well-connected, suggesting that these appointments had little to do with skill or virtue and a lot with ancestry and greasing the right wheels. Most likely the polemarchs were appointed by the ephors (who called up the army); it is possible that, like the tribal commanders at Athens, they chose the officers below them, who chose the officers below them, and so on.
If this is right, then no matter how political the appointment of polemarchs was, there would have been considerable room for promotion on merit in the less prominent ranks. As long as you did not aspire to commanding your own army, you could probably rise from an ordinary hoplite to as high as a lochagos through the abilities and respect you commanded. If you were known to be an outstanding athlete or speaker in your year-group before you came of age for military service, that would probably help to get you on the ladder early.
In the Spartan case, then, there was a more elaborate rank structure, which amazed and confused non-Spartan Greeks. Unlike at Athens, none of these ranks were filled by election. While you had no hope of ever becoming a king and leading the full Spartan levy, it may have been possible to rise to a mid-level rank based on your performance in your education or in the course of several campaigns.
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