I think you have the wrong view of how Greece worked in those days.

Greece was divided into a multitude of city states, some of the allied in Legues, but not all. These city states were spread from the Crimea to Iberia, with Sicily and southern Italy (Magna Graecia, or larger Greece) as prominent places.

Some of these city states were oligarchies, were a few powerful families ruled. Some were tyrannies (which simply meant a single ruler back then) and a scant few others were democracies. A few others were monarchies.

Those that were democracies did not have a King and thus no tasks for him. Athens, the most prominent democracy, would probably be considered a failing democracy under today's standard - citizens could vote other citizens out of their citizenship and voting rights (which happened on a massive scale on at least one occasion).

The two Kings of Sparta were not Kings in the meaning we give the name now - they were more like military commanders, and they were appointed by the Spartan assembly, which included all Spartan citizens (who were a very small minority of Spartan society), so Sparta can probably be called more of an oligarchy than a monarchy or democracy.

Macedon was a Monarchy, but was not a city state and was during the classical era considered only semi-Greek by the other Greeks.

Argos, Thebes and Corinth all developed from oligachies into monarchies during the era though.

But to answer your question - there were no democracies in the Greek world that also had a monarch.

As /u/vonadler says, there was no unified "Greece" with a singular government, but there are still some common processes that the Greek city states went through in their political development. These often had different results in different areas, but they can often be seen as responses to a similar set of social circumstances.

Pinning down kingship in Greek history is somewhat complicated because by the time real written sources begin the institution had either disappeared or been completely changed throughout the Greek world. It left a powerful memory on Greek mythology and literature, but by the sixth and fifth century "kings" as such can only really be found on the margins, such as in Macedon or Epirus. To find it in the core areas of Greece requires leaping back to the Bronze Age, in which two relevant institutions appear in the Linear B tablets: the wanax, which most closely corresponds to a conception of "king" as found in the Near East, and a basileus, which seems to have been a sort of subordinate chief. The word wanax remained in the Greek language as an intentional archaism when referring to kingship, but the institution itself seems to have disappeared with the Mycenaean palace complex in the turmoil of the so-called "bronze Age Collapse"--or at the very least evidence for it disappears with the loss of the practice of Linear B writing and the disruption in the artefactual record. The term basileus, however, survived, and became the common word for king used in, for example, the Homeric epics. It is possible that the basilei were local leaders under the supra-regional wanaces, and with the destruction of the palace society they were able to solidify their autonomous authority, but this is speculative.

The problem with this theory is that we don't actually have direct evidence for the basilei. They dominate the landscape of Greek mythology, but we simply cannot take the dramatic scene of the Antigone or Agamemnon as an accurate depiction of the politics of an ancient Greek city, as that was neither within the intention or capability of the playwrights. Homer is sometimes taken as a source, but the society of Homer is deliberately archaic, he was not portraying the society of his own day but of a remote and idealized past. It is possible that the basilei held on for a few centuries after the destruction of Mycenaean palace society, leaving a heavy ideological footprint on the oligarchic societies that followed, but we simply don't know. Unlike the Romans, the Greeks did not really maintain much in the way of a concrete memory of a transition from a regal to non regal political order.

The society described in the early literary record can be broadly described as oligarchic, with elite families competing for influence among themselves. The social processes of the Archaic, particularly colonization, economic and demographic expansion, and the development of the classical polis society, had different results in different areas. In Athens, democratic forces "won", and a society emerged that carefully curtailed the competition of elites (for example, through the practice of ostracism, in which prominent citizens who got too big for their britches would be kicked out of the city for a decade). In Sparta, the oligarchic forces closed ranks, created a heavily regimented society carefully delineated along class boundaries. However, neither the basileus nor, and I find this crucial, the fear of a basileus survived. The Greeks expressed political angst through a fear of a tyrannos, which was monarchical but explicitly not basilei, as by definition they come to power through "illegitimate" means.

In short, in order to find "kings" as such one needs to go back to the Mycenaean period, and by the time the classical literary record kicks in they were long dissapeared and their memory filtered through many layers of literary artifice. Unlike the Romans who explicitly described a process of the loss of kingship, the Greeks (or at least the Athenians) did not retain a memory of throwing off the institution.

I think this is given a very good description in Pomeroy's Ancient Greece.