OK, this question calls for a multi-faceted approach, so I'm going to start with a very brief overview of the anthropological perspective before moving on to some actual history. While anthropology can't really reveal where and when a phenomenon originated, it does provide perspective on a more fundamental matter that has to be considered before we can get to the matter of origins - what exactly is a king, and what distinguishes them from someone who is simply in charge?

Anthropology, ever since Frazer's Golden Bough, has tended to emphasise the ritual and cosmological significance of what is called "divine" or "sacred" kingship. The classic anthropological example of this is the much-discussed Shilluk Kingdom in southern Sudan, which is, essentially, a non-state: there is a king, but no administrative apparatus; it's simply a group of settlements that are unified by ceremonial customs centered around the reth, the Shilluk divine king who is considered to be the reincarnation of the culture hero Nyikang. The Shilluk king does not really rule in the conventional sense, because he by and large has no means by which to rule, but at the same time his existence is essential for social order among the Shilluk and the aristocracy who hold power in individual village. Basically, the figure of the king makes possible the unity and the continuity of the Shilluk society, and the king, as Nyikang, is theoretically immortal.

So, what initially distinguishes such kings from other types of rulers is that their existence is crucial for society to sustain itself. This goes with the often quoted statement by A.M. Hocart, "The first kings must have been dead kings." (Social Origins, 1954) According to his theory, kingdoms are initially ceremonial and religious organisations which, being centered on ghosts, spirits or gods, develop institutions that perpetuate themselves over generations. In the process, almost inadvertently, they also start to take on administrative functions centered around specific individuals who serve as heads of these cults. In comparison, the so-called "Big Man" of egalitarian tribal societies might have considerable societal power due to his personal ability to negotiate and coerce others to do things, but social order doesn't necessarily fall apart in the absence of such a chieftain, nor does the role inevitably pass down to others if there is no sufficiently gifted individual to become a new Big Man.

The actual origin of kingship is not a process that can be attested historically, since the earliest kings probably emerged before substantial literature, and archaeological sources only tell so much. However, the general scheme of ceremonial kingship fits reasonably well with what is known about kingship in ancient Egypt and Mesopotamia. Probably the earliest unambiguously attested kings are the rulers of protodynastic Egypt in the late 4th millenium BCE, but I am not an expert on the Egyptians; it should suffice to say that Egyptian kingship was explicitly divine in character, with the entire Egyptian society revolving around the pharaoh, who himself was essentially deified.

What we know about pre-dynastic Sumer might be more interesting, though there's no consensus on how the period should be interpreted. While Sumer was divided into city-states, it seems that temples were clearly the predominant administrative organisation that owned land, organised work and redistributed resources; these temples, in turn, had a long continuity, with the earliest village temple of Eridu built during the 6th millenium BCE. During the third millenium, when historical sources are present, the rulers of these city states did not always have the title of lugal, the Sumerian word for king, but rather used the titles ensi or en. The precise interpretation of these terms remains contested, but in some cases, it appears to be the case that the the god of the city-state has the title of lugal, while the mortal ruler of the city-state uses the lesser, subservient title of ensi, to the effect that the ensi rules as a governor in the name of the god. This is particularly evident in the inscriptions of Gudea, who appears to have deliberately eschewed the use of the title lugal as a sign of piety.

Kingship in Mesopotamia, in any event, was considered to have had a divine origin, and even when mortals used the title, it effectively meant participation in a primeval institution originating with the gods. Even as the institution of kingship gradually eclipsed the temples in power and wealth, and the title ensi eventually fades into obscurity while lugal, and its Akkadian equivalent, sharru, become the sole titles of rulers, kingship remained essentially divine in character and something appointed by gods.

I don't know what kind of answers you might get from historians, but you could always try archaeologists or anthropologists. It would be interesting to learn if there are any clues to this in prehistoric archaeology. (Do we have Ask Archaeologists or Ask Anthropologists subreddits?) Many of our primate cousins have 'Alpha' leaders in their troops, so could the concept of 'king' come from humanity's very distant past?

I'm reading about bronze age Crete and Aegean right now [1]. First signs of a stratified society in this area are graves that differ in amount of prestige goods. This is long before any texts were written. Elaborate burials that indicate a possible difference in status can be seen in ice-age graves, too, as old as 30'000 years. [2].

Written history starts in different independent areas (China, Middle-east, South-america). When you say "king" I assume you also imply a state structure? Before kings, there might have been "big men" with power. But there's a big difference between "big men" and a fully functional, administrative state with a king (or priest-king).

If you ask about the origin of status differences in general among human social systems, then yes, anthropologists might be better.

Edit: About

Does written history "start" with this structure already in place?

First written records are administrative, about distribution of goods and work which assumes a central power, so yes (no source, see Sumerian culture).

[1] Shelmerdine 2008: The Cambridge companion to the Aegean bronze age.

[2] Jill Cook, 2013: Ice age art, arrival of the modern mind