How was the divinity of Roman emperors played up in contemporary media?

So, I figured I'd finally write an answer. What I get that you are asking about is essentially the propaganda of the Imperial Cult. I've seen that at least one other flair here refrains from using the word propaganda because it really is a modern term, but I prefer to use it because it accurately describes the phenomenon we are talking about. This narrows down the propaganda quite a bit from most questions I've answered around the same topic on the subreddit, since you're asking about the cult specifically. I'm not all too familiar with the event you're referring to, so I can't comment specifically on it, but the way it's written is along the lines of the usual propaganda of Augustus as the ideal citizen. If you havn't, you may be interested in reading Augustus' Res Gestae, because although not all of it is related to some sort of divinity, particular parts of it are (specifically chapters 4,7,9,10,11,12,19,21,24).

First off, it would help you if you took a glance at this post that I wrote a while ago. In it I describe the different kinds of Imperial Cult. It's helpful, because how the divinity would be played up would differ greatly depending on where in the empire you were - not to mention that it also differs over time. During Augustus, before any emperors had been deified at all, there was a temple to the divine Julius in Rome, but out in the provinces such things were scarce (off the top of my head I can't think of any). Altars on the other hand were quite wide-spread during this time. One of my favourites is an altar to Gaius and Lucius Caesar that's been found in Lusitania (roughly modern Portugal) which in effect shows how Augustus were grooming them as successors. Another altar, an older one, has been found at The Three Gauls, a place in modern France, which was dedicated by Drusus (Tiberius' brother). This particular altar is highly interesting to us because it shows that other members of the imperial family worked to legitimate the family's claim to power. It isn't an altar by Augustus for Augustus, but by the son of the emperor's wife for Augustus.

However, this is just the situation in the western part of the empire during this time. The eastern parts had a completely different tradition (and understanding) of what ruler cult implied, which meant that while Augustus was strongly against being worshipped as a god in the west, he couldn't forbid it in the east. Therefore he allowed the erection of temples in the east but not so in the west. Altars were erected for the living, while temples were reserved for gods. Here's a quote that describes how Tiberius later handled a situation where a western province sent a request to erect a temple to Tiberius and Livia (who was not yet deified):

[---] I am aware, Senators, that many deplore my want of firmness in not having opposed a similar recent petition from the cities of Asia. I will therefore both explain the grounds of my previous silence and my intentions for the future. Inasmuch as the Divine Augustus did not forbid the founding of a temple at Pergamos to himself and to the city of Rome, I who respect as law all his actions and sayings, have the more readily followed a precedent once approved, seeing that with the worship of myself was linked an expression of reverence towards the Senate. But though it may be pardonable to have allowed this once, it would be a vain and arrogant thing to receive the sacred honour of images representing the divine throughout all the provinces, and the homage paid to Augustus will disappear if it is vulgarised by indiscriminate flattery. (Tac. Ann. 4.37)

Interestingly enough, despite what I wrote above, there is a temple in Britannia to Divus Claudius (the 4th emperor) that undoubtedly was erected during the emperors' lifetime. This particular temple has sprung some controversy in the field because it essentially works against what we have accepted as fact - that temples weren't erected until after the emperor was deified. Generally it is tentatively explained that this happened because Britannia was such a distant province which allowed for slightly different customs. A bit of an unsatisfactory explaination, but considering that Tacitus and Suetonius, our main sources on the subject, offer different accounts on Claudius' deification (which was in several stages) it is less surprising that such a thing could happen.

These practices seems to have slowed down a bit during the course of the first century AD, because the amount of finds is a lot more during the Augustean period than during the rest of the Julio-Claudian era. However, with Vespasian's reign the practice seemingly is reborn - likely because Vespasian understood what Augustus had done and copied it - and during his reign there are a lot of new stuff, and his impact also shows after his death with numerous inscriptions over priests who have been sacerdos divus Vespasianus or flamen divi Vespasiani (the two different priesthoods associated with the Imperial Cult).

The impact of the Imperial Cult dies in large after "the good emperors" (Nerva-Marcus Aurelius) and the army becomes the new way to display power - an overall change in the empire that's unrelated to this subject. Anyway, jumping forward about 70 years from Marcus Aurelius' reign, there is a certain decree that shows how the view of the Roman emperor's divinity has changed in the empire. That decree is essentially the last really impactful aspect of the Imperial Cult in that the emperor Trajan Decius through this decree kicked off the first empire-wide persecutions of Christians. The decree required every inhabitant (ie. not just citizens) to sacrifice to the gods. What made this decree significant wasn't that it was empire-wide, such decrees had been made before. This decree however worked down to an individual level and thus had a much larger impact on society than earlier, similar, imperial decrees. Ironically this decree also signifies the beginning of the end for the Imperial Cult; by Constantine's reign 50 years later it was dead.

Sources:

For excellent maps detailing these altars and temples, see Étienne (1958), *Le Culte Impérial dans la Péninsule Ibérique d'Auguste à Dioclétien.

For the cult in general, I strongly recommend Fishwick's work The Imperial Cult in the Latin West, Studies in the Ruler Cult of the Western Provinces of the Roman Empire (2002, 2004), parts 1-3.

Concerning the deification of Claudius, see Fishwick (2002), "The Deification of Claudius" in Classical Quarterly, 52.1, pp. 341-349.

And for the mentioned decree, see J.B.Rives (1999), "The Decree of Decius and the Religion of Empire" in Journal of Roman Studies, vol.89, pp. 135-154.