Factors from my observations and opinion.
The obvious is traits and retinue that either increase or decrease popularity. The death of a faction leader can have a significant impact on these factors, all due to retinue.
Your strength compared to your Roman allies strength, I think it factors in your overall ranking with the other Roman factions.
Senate missions, hard to pinpoint the factors here but it appears too many succesful missions in a row has a huge impact. It appears accomplishing 3/4 on average keeps the senate happy and keeps enough failures in your name that the plebs don't honour you enough.
Canceling trade rights has a huge impact and provokes other Romans to attack you and possibly getting them in more trouble than you. Ah yes politics
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