We have a diary of Emperor Marcus Aurelius Antonius Augustus in which he frequently meditates on his position in the universe. In it, at times he refers to himself as essentially being a thief, but on a greater scale (referring to his waging of war) and never refers to himself as having what we today would refer to as divine powers. As an Imperator, Marcus Aurelius was the focus of the imperial cult, however, unlike his son Commodus in his life Marcus Aurelius never directly pressed the issue of identifying himself with a single divine figure (as his son did with Hercules). In his meditations, Marcus Aurelius does appear to believe in a divine order to the universe and that he has a special place within it. At one point he says:

"[If any have offended against thee, consider first]: What is my relation to men, and that we are made for one another; and in another respect I was made to be set over them, as a ram over the flock or a bull over the herd. But examine the matter from first principles, from this. If all things are not mere atoms, it is nature which orders all things: if this is so, the inferior things exist for the sake of the superior, and these for the sake of one another"

Does this belief in his own special place in the universe mean that he believed in his own divinity? Or is that just the sort of thought that any man who was doing fairly well as the leader of the most powerful nation in the known world might think, if he let it get to his head? What does it mean to be divine?

Certainly, it doesn't appear as though he thought much of any ability of his to answer prayers or benefit from the burning of incense to shrines of the imperial cult. However, I think the best answer to your question, in terms of this one emperor in this one system of Imperial Cult, is that it's complicated.

If we look at the example of Alexander the Great and ruler-cult, it seems unlikely that rulers would have truly believed this. In the case of Alexander, we can glean a lot from his coinage. He's often portrayed differently on coinage depending which region the coin comes from. He's frequently portrayed as Heracles-- many classical historians attribute this to an attempt to legitimize his authority in Greece as a Macedonian. Elsewhere, he's depicted on coins as a descent of other gods from other pantheons. To say that he truly believed himself to be descendant from these deities is unlikely, as there is strong evidence for Alexander's opportunism. Although not necessarily an attempt to portray himself as a deity, ruler cult can also be seen in the life of Julius Caesar. His permanent wearing of the crown of laurels, his purple robes and the construction of his "throne" in the senate house were all attempts to create and legitimize his authority as a king without being named as such. Imagery is very important in the ruler ethos and there is a lot of evidence for rulers using this imagery so solidify and consolidate authority. Portraying oneself as a god or as a descendant of a god worked to this effect and helped those like Alexander to consolidate their authority (of course, the question of whether or not Alexander truly consolidated his authority throughout his empire is an entirely different issue).

There is a record by Bernal Diaz of the interaction between Montezuma and Cortez, speaking through Cortez's Aztec interpreter Malinche clearly stating that the Aztec King Montezuma did not believe himself to be a god.

The following is from Diaz's The Conquest of New Spain (1963).

"...his vassals...had told him we shot out flashes of lightning and killed many Indians and that we were angry Teules, and other such childish stories. But now that he had seen us, he knew that we were of flesh and blood..."

"Montezuma replied with a laugh...'Malinche, I know that these people of Tlascala with whom you are so friendly have told you that I am a sort of god or Teule, and keep nothing in any of my houses that is not made of gold or silver and precious stones. But know you are too intelligent to believe this and will take it as a joke. See, now, Malinche, my body is made of flesh and blood like yours, and my houses and palaces are made of stone, wood and plaster. It is true that I am a great king, and have inherited the riches of my ancestors, but the lies and nonsense you have heard of us are not true. You must take them as a joke, as I take the story of your thunder and lightnings."

Bernal Diaz in The Conquest of New Spain (1963)

I can answer this in relation to Alexander and the Successor Kingdoms. Particularly in the Successor Kingdoms, the rise of ruler cult was primarily political. It progressed from cities voting divine honours in gratitude for the actions of the king, for example the Scepsis decree honouring Antigonus, or to play one Successor against another, for example where the city of Rhodes deified one of the Ptolemies ostensibly as a slight to Demetrius. The citizenries voting these honours understood that they did not have the cosmic power to assign divinity but they voted divine honours in recognition of the fact that 'the other gods are either far away, or they do not have ears, or they do not exist, or do not take any notice of us, but you we can see present here' (Ithyphallic hymn in Athens, recorded by Athenaeus).

As time went by, Successor kings themselves started requiring or enforcing their pseudo divine status more rigidly, especially in Egypt where the practice was known from the Pharaonic period. Ultimately though, there's little agreement on where exactly ruler cult fit into the schema of state religion. It's my historical opinion that coming as it did from such a political base, only a deluded ruler could believe in his own divinity without complication.

It wouldn't be that they were unaware of their humanness and truly believed they were divine but rather I would think that they had a sense of some special purpose that they rightfully, in their opinion, possessed or were rightfully guided.

I don't know if this can count as Divine King or not, but Chinese rulers do think of themselves as the true holder of Mandate of Heaven who were given the rights to rule 'all under the Heaven' (天下) that they often had the title 'Son of Heaven' attached.