What did the very first king on Earth do to become king?

That happened in prehistoric times, so all we have to go by is early Uruk and Sumerian inscriptions such as the Sumerian King List. It claims that the kingship descended from heaven, and that the first king was Alulim, who ruled the rule of the city of Eridug, and who the manuscript claims lived and ruled for 28,800 years. And it goes on to claim that 7 more kings ruled for a combined total of 208,800 years in Eridug, before a flood devastated the city. No, it wasn’t Noah’s flood. Archeological evidence shows it was a localized event. There is no evidence of any flood ever covering the top of all Earth’s mountains.

Further, the claims of such extreme age are highly unlikely to be factual, as radiocarbon dating of the manuscript making these claims places it in the Early Bronze Age around 2900 BCE. We have no archeological finds verifying that the Antediluvian kings listed ever really existed. They may have, though they almost certainly didn’t live tens of thousands of years. They may also be entirely mythological.

Gilgamesh is the first king we have much substantive evidence of. He was the 5th king of Uruk, and so probably descended from the 4th king. Enough artifacts and cuneiform inscriptions have been found to suggest he actually existed, however the claim that he ruled for 128 years seems highly suspect. It was common in those times to make grandiose, superhuman claims for revered leaders.

How did the first great leader grab power? Knowing human nature, a good guess is by playing a very deadly adult version of the childhood game, King of the Mountain. He was probably the biggest, baddest war lord in his corner of the world, and seized power by eliminating all opposition.

Think flood plains and hydrology.

If your family and friends controlled a particularly fertile area around a flood plain, eventually people are gonna want to settle close and offer you a share of their crops to let them farm without trouble. It snowballs after a few generations. The first tenants that came in are going to look to you to keep undesirables out of their lands, and so on, and so on, until you have a complicated alliance looking to you to keep out new folks, or at least tax them heavily to stay around.

“The Lady of the Lake, her arm clad in the purest shimmering samite, held aloft Excalibur from the bosom of the water signifying by Divine Providence that I, Arthur, was to carry Excalibur. That is why I am your king!”

Usually one takes over by force.

Offer safety in return for loyalty.

Stepped forward as a leader of the people.

He found people simple enough to take his claims to power seriously.