What a Homeric king is
The word is basileus, and it does not mean what "king" means in later European usage. The Odyssey's Ithaca contains several men called basileus; Telemachus says so himself, remarking that there are many of them, young and old, and that one of them may well hold the kingship now his father is gone.
A Homeric basileus is the head of a leading household who exercises authority by a combination of birth, wealth, personal capacity, and the willingness of others to be led. It is not an office with fixed powers, and it is not straightforwardly hereditary.
This matters for the plot. Telemachus's claim is real but weak, and the suitors' assumption that whoever marries Penelope has a claim on the kingship is not absurd within the system.
How authority is actually exercised
By speaking in assembly and being listened to; by feasting others and giving gifts; by fighting well; by having a functioning household. The poem shows all of these and shows Telemachus acquiring them in order.
What it does not show is machinery. There is no standing army, no bureaucracy, no scribes, no taxation, and no means of compelling obedience without persuading. This is why the assembly in Book 2 fails: the mechanism for making a decision exists, and the mechanism for enforcing one does not.
The king as guarantor
The poem's positive statement about kingship is put in Odysseus's own mouth, to Penelope in Book 19: the reputation of a blameless king who upholds good order is like that of the earth under him — the land bears wheat and barley, the trees are heavy with fruit, the flocks breed, the sea gives fish, and the people prosper.
That is not a theory of legitimacy. It is a theory of consequence: good rule shows up as fertility and prosperity, and bad rule as their absence. Greek political thought kept returning to this.
What kingship is not, in this poem
It is not divine right. Kings are said to hold their honour from Zeus, but the poem contains no notion that they may not be removed.
It is not absolute. Alcinous consults his elders. Agamemnon in the Iliad has to be argued with, and loses the argument.
And it is not secure. The Odyssey's whole situation is a demonstration that a kingship left vacant for twenty years simply dissolves into the competition of the leading households.
The gap with the archaeological record
This is important and often missed. The society of the Homeric poems does not match what the Linear B tablets show of Mycenaean Greece.
The tablets describe centralised palace economies with a wanax at the head, literate scribal bureaucracies, detailed inventories of land, labour, textiles and offerings, and a hierarchy of officials. Homer's world has none of that: no writing, no bureaucracy, no palace redistribution, and authority held personally rather than administratively.
Moses Finley argued in The World of Odysseus (1954) that the poems therefore reflect a later, simpler society — broadly the Early Iron Age — rather than the Bronze Age they are nominally set in. The dating remains disputed; others hold the picture is a poetic construction that matched no single period.
Either way, the poem should not be used as a description of Mycenaean kingship. See The Odyssey and Mycenaean Greece.
Later interpretation
Aristotle's Politics discusses Homeric kingship directly as one of his types of monarchy — generalship held for life, with limited powers, by consent. His reading is the beginning of the political-theory reception of the poems, and it is closer to the text than most of what followed.