ELITE Roman culture was heavily influenced by Greek. We know even that most conservative Roman Cato the Elder knew Greek and studied Greek literature. Quintilian preferred if possible his students start Greek before Latin. Studying oratory in Athens or Rhodes was common for young aristocratic men finishing up their education.

How far down the social order did that kind of familiarity with Greek penetrate? Hard to say. We would only know from what writing survives, and there's not a lot of lower class writing left. Tombstone inscriptions, graffiti, stuff like that. It's not representative; it's just what happens to survive. On top of that, estimating the level of literacy in any language among these lower classes is tough.

There was no conscious attempt to force assimilation to Latin. Romans generally let people carry on as they were, and in fact Greek was always the dominant language in the East. Alexander had spread it throughout the old Persian Empire and it took at least at the administrative level and the urban centers. Latin never made much headway against Greek. The text of the Res Gestae preserved in Ankara exists in both Greek and Latin, for instance, and Claudius' Letter to the Alexandrians only exists in Greek.

In the West Latin was more effective, but not out of any policy of assimilation. I'm not sure anyone has a widely accepted explanation for this. We know that the West became much more urbanized under Roman rule than earlier, and that involved adopting Roman institutions like fora, temples of the imperial cult, the establishment of military coloniae and such (how exactly this worked is hotly debated, ranging from Rome imposing Roman institutions on the provinces, to provincials wanting to participate in Roman culture, to Roman stuff being a thin veneer on top of local stuff). Latin would have come along with public monuments if nothing else.