I suppose it's a question about ancient court politics and will vary depending on who you choose. Since I was reading about Cleopatra, who was stylized in the manner of Isis, I have to wonder how much political gain there was to be had in this posturing. No one vying for power could walk into court one day and say, "By the way, I'm a God." They would get laughed at by the other nobles, right? No smart general would choose to side with someone who pulls such a stunt, but it seems to be very common in ancient history, and can even be seen to an extent now in North Korea with the 'deification' of the Kims.
Basically, what was the MO of these claims? Did the titles and accolades have to be thrust upon these people by the peasants, or was an alliance with the clergy something of a necessity? How did the claim of divinity evolve, as well as the public perception of it? How many rulers did this end up undoing because it 'went to their head' and they developed a superiority complex?
Is there a pattern? When a person claims divinity does it tend to follow a archetype? What I mean by that is this; does the person first have to come into some modicum of power, followed by some major military or political success, with rumors evolving around them which they come to embrace?
I apologize if those post is too open ended with the guidelines, but I am not a historian so I don't entirely know how to pose good historical questions.
I always feel that trying to make a general rule for something like this is somewhat a fool's errand: there are too many examples and counter examples for it to be coherent, concise and accurate. So, if you don't mind, I'll use a few specific examples.
For the purposes of this explanation, I'm going to classify something like the king's touch as a divine link: people saw it as evidence that the king had a divine mandate to rule. How did kings get the right to perform the king's touch? In England and France, it was after they were crowned and anointed. The coronation ceremony, which involved the clergy and nobility ceremonially 'electing' the heir of the previous king onto the throne was therefore how their divine/ sacerdotal status was affirmed. In real terms, this did not really help them politically: of course, it legitimized their rule etc, but it did not actually shift power dynamics. Power was personal; some anointed kings were able to use their position like a ball and chain, accruing lots of power. Others, like Stephen of England or Henri III of France were dominated by those around them. You say that no smart general would choose to side with someone who pulls such a stunt, but that depends on the situation. Charles de Bourbon's rebellion in 1523 fell flat because his armies/ vassals chose to side with the divinely anointed Francis I instead. They felt that the anointed king had more legitimacy than a rebel. On the other hand, we see anointed kings kicked about, like Edward II of England whose deposition was passed in Parliament. I hope that addresses the questions in your first paragraph.
Peasants would not have the authority, on the whole, to thrust these claims on the monarch, it was almost always the clergy and the nobility that affirmed and conferred this status. Public perception of it is harder to gauge: we know that in 1649, when Charles I was executed, people dipped handkerchiefs in his blood because they felt that it carried the king's touch. Someone more knowledgeable on the Enlightenment may be able to explain this to you. People in France clearly still saw the power of the holy oil of anointment, as the holy ampulla within which it was carried was smashed in the Revolution. Rulers like Charles I were to an extent undone because of his intransigence, perhaps coming from his belief in his own semi divine status.
I hope this helps!