How did most kings in ancient Greece come to power?

They moved into Europe as nomands, looting the peoples who had previously settled there. When there was nothing left to loot they had to settle down and grow their own food and wealth. The established city-statesbuilt around a fortress, then a city. These independent city-states became powerful in their own right, and often combined into leagues for mutual defence.

They also spread their surplus populations around the Mediterranean and Black Seas, eatablishing about 2000 city-states, which might have been all-powerful in the region but for their habitual warring amongst each other, which so weakened them that they presented an easy target for first Macedonia, and then later Rome.