That was why I asked about Tenotchitlan-keeping them strong(er) might ease the repeated political vacuums of the Maya lands and maybe even putting them in a position to become undisputed masters of Mesoamerica. As for Kingship, a general summary would be more helpful since I don't know enough to have good specific questions.
Don't you mean Teotihuacan?

In any case, I'm personally a bit dubious they can be the undisputed masters of Mesoamerica for several more centuries. They were already the undisputed masters before they collapsed, or at least during the Early Classic period when their influence over the Maya region was the most obvious. Though the continued survival of the city and nation itself should help with stability in any case, even if they don't remain the massive all-powerful empire they were at their height.
As for kingship, where to start? I suppose the native etymologies of their words shall do to give you an idea of how they were thought of. The typical word used for kings in Mexico (as in, the western part of Mesoamerica beyond Chiapas) was "Speaker", most exemplified by the Nahuatl word "Tlatoani". It's also been theorized that the Maya word for lord, "ajaw", has roughly the same meaning though this has not been confirmed. The king was someone who ruled (ideally) through charisma and willpower in addition to their own sacred qualities. That is to say, Mesoamerican kings weren't quite conceived of as the cliche god on earth type of king, but they still had a sort of sacred quality about them. Ideally they were also a proactive sort, who took action and used a combination of strength, intelligence, wit, and even guile. The Maya exemplified the qualities of the ideal lords in the story of the "Hero Twins", most famously told in the K'iche' Maya's Popol Vuh. The twins Hunahpu and Xbalanque set out to avenge their father (Hun Hunahpu) who was cruelly slain by the Lords of the Underworld. In the story the twins are not above using a little trickery in addition to their skills in order to gain their vengeance, and are seen as pure heroes. While the Popol Vuh doesn't necessarily reflect Classic-era beliefs (the primary difference being that the Classic Maya placed far more emphasis on the death and resurrection of their father the corn god) they still revered one of the twins in particular as a sort of patron or epitome (I know there's a better English word I'm forgetting) of kingship, Hunahpu to be precise, or Hun Ajaw in Classic Ch'ol. As you might tell from the name, his status is reflected in his name. Kings themselves also placed a lot of emphasis on descent and lineage, and it was common if not the norm for dynasties to trace their origins to mythical beings. Being a hereditary monarchy, ancestry and legitimacy were quite important. Tracing themselves to mythical people also gave their sacred status more credence, as they were also the highest religious officials in addition to being the heads of state, and as such they had a special link to the gods. Their very presence was also important and helpful in a way. The language used to describe a king's actions also takes on the same sort of description as that used for normal everyday activities. This reflected the idea that kings were patrons for their people in a way, helping their kingdoms grow, giving sustenance, etc. The king was seen as a sort of protective agent this way. In a more superstitious sense, they were also capable of sorcery and could use their spiritual guides/assistants (nagual in Nahuatl, waay or uay in Mayan) to bewitch enemies or traitors and the like. More practically, kings were the penultimate power within their own domains and the absolute rulers, though they could still have to take their own subject lords and the Popol Nah (council houses) into account. They were owed both tribute and loyalty. They might be subservient to more powerful kings and emperors, but they still had to be afforded some respect even by them. In courts, the rules for showing respect to superior lords had to be obeyed stringently, as not to offend, which is obviously a big deal back in those days. In all likelihood, even a weak king who simply reigned while they had regents doing the ruling still would have to be afforded the same courtesies. When they ascended the throne, the change was reflected by the new king's adoption of a regnal name, which I'd assume reflected the new status afforded to them rather than that of a still powerful but not as sacred or holy prince, though in practice many kings seem to have preferred going by their personal names. Taking the throne could take place quite some time after the previous tenant died. It was probably not a matter of simply grabbing the crown and plopping it on the heir's head. There were rules and rituals to be obeyed. And once one was king, they had to be present at the major rituals and ceremonies of their kingdom. When wars were commenced, they were ideally leading their armies though this was probably not always done. If one was captured in battle however, they were bound to either become a puppet ruler or a sacrifice. If the latter, then it would be important that they died with dignity. In death kings were luckier than all, as they were guaranteed a place in paradise with their ancestors by their status.