His father was a Roman, Cassius Apronianus, a senator and imperial governor of Ciicia and Dalmatia. Thus by blood he was "Italic," a member of the ancient Cassia clan. His mother may have been the daughter of sister of the Greek thinker called Dio Chrysostom, from whom he might have taken the honorific name "Dio." This word is not part of his Roman name officially. He might have also had the name Claudius (unlikely if you ask me) or Cocceianus. His actual Roman name was probably [praenomen] Cassius Dio Cocceianus. In Greek practice, the Dio tends to come first (Δίων ὁ Κάσσιος), while in Roman practice, it would come second (treated as a cognomen). Thus two millennia of confusion about this guy's actual name, between Dio Cassius and Cassius Dio. Either is acceptable.

So his father was Italic Roman, but his mother was possibly of Greek descent. And he was born at Nicaea in Bithynia, which is more "Greek" than "Roman" culturally. Its inhabitants spoke common Greek, among other languages, observed a typical Romano-Hellenistic culture, but were administratively a province of the Roman empire. So he was part Roman, part Greek (possibly), and born in a mostly Greek-speaking Roman province. He wrote in Greek by choice, not necessity, and he was of course fluent in both Greek and Latin, just as were the majority of Roman aristocrats at this time.