Speaking to one example, as there were quite a few Roman emperors and I'm sure their personal beliefs differed, we have the benefit of having the Meditations of Marcus Aurelius available to us. The text was composed in Greek (usually titled as Τὰ εἰς ἑαυτόν or essentially "writings to himself), and was intended for personal reflection and self-improvement rather than publication, so it seems fair to expect them to be about as honest as we can hope to get -- akin to having a chance to read the private journals of US presidents so that we can argue about their religious beliefs.

Now, a particular quote comes easily to mind from the second book of the Meditations because it's often garbled as a quote that shows up in atheist/skeptical memes in a way that suggests that Aurelius doubted the existence of the gods, but the original reads quite differently:

But to go away from among men, if there are gods, is not a thing to be afraid of, for the gods will not involve thee in evil; but if indeed they do not exist, or if they have no concern about human affairs, what is it to me to live in a universe devoid of gods or devoid of Providence? But in truth they do exist, and they do care for human things, and they have put all the means in man's power to enable him not to fall into real evils.

Emphasis mine. Again, we're looking at one of the more intentionally spiritually/philosophically minded emperors, and no source is perfect, but I simply don't know what better source we could want into an ancient ruler's private thoughts than the Meditations.