I believe that this debate misses part of the truth, as most modern debates about Caesar do, because it allows itself to be misled by the modern meaning of the words "king" and "republic".
In our time, these words have a purely political meaning while in Rome, at the time of Caesar, they were as much religious as they were political. Implicitly, they asked the question: "Can a man (especially a leader of men) become a God?" The Greek answer to this question was an emphatic "yes", while the Roman traditional, republican, answer was an equally emphatic "no".
The traditional Roman answer was related to the strong standards of piety Rome liked to consider its own. Allowing a man to claim godhood was to challenge the gods and they would not like it. By contrast, the Greeks had always made it relatively easy for men to become gods (the hero path) and this tendency had become even more pronounced after Alexander, with almost every ruler of some importance being granted divine honours.
The problem of the late Roman republic is that after the conquest of the Greek east, and the influence of the Scipio, the Romans had come to consider the Greeks as their cultural masters. Some, like the Cato family, deeply disliked the Greek way and tried to resist this trend. But it was a losing battle. The bottom line was that, by Caesar's time, the populace was more than ready to pay divine honours to a successful military leader. And this was especially true of soldiers who, through campaining in the East, had been most exposed to Greek customs.
Caesar, like Pompey before him, had campaigned in the East as well and had therefore had to interact on a regular basis with semi-divine rulers who paid him homage because he was Rome's representative, of which they were vassals. Yet, he was supposed to remain a man when he was in Rome.
We will never know what Caesar really thought because the only text we have from him (De bello Gallico) was a propaganda piece, and maybe ghostwritten, and the opinions of others about him are of course slanted. One thing is sure, though. When, after his death, Caesar was deified on the orders of Octavian, the cult was wildly popular. Caesar really did become a god and this is why we still use the words Kaiser and Tsar today.
Could Caesar have mentally resisted the temptation to think of himself as a superior being, while everything seemd to be yielding to him around the whole world? Is it realistic to think so given what we know of human nature? I think not.
It is therefore likely that Caesar was just torn between 2 irreconcilable positions. He was a god in his own eyes but he could not acknowledge it while he was in Rome. So he tried to muddle through, as his behaviour shows. It seems to me that he did not have a precise plan regarding how to legitimize his own power, and neither did he have one regarding his succession. He improvised and took whatever decision seemed judicious based on his gut feeling in whatever situation he found himself in. The campaign against Parthia was the next logical step. Maybe he thought that the Roman people would acknowledge him as a god at last, if he matched Alexander.
It is with Octavian that we find someone with a plan. As a young man, he had witnessed his uncle's trajectory and no doubt pondered on how to solve the dilemma he faced. By contrast with Caesar, Octavian had hindsight, and had had time to reflect when his turn came to face the same choices. We know what his plan was, and it succeeded. It was as much religious a religious reform, a thorough reorganization of Roman state religion, as it was a political settlement.