He spoke Classical Latin, but in his time the use of that language was mainly limited to politics and religious ceremonies. And as an army commander he must have picked up a lot of Vulgar Latin (sometimes called "colloquial Latin"), the language spoken on the street by commoners.
But as a patrician he would have spoken mainly Classical Greek with his equals. Since the Roman annexation of Greece, the Greek language gradually obtained a unique place in the Roman world, owing initially to the large number of Greek slaves in Roman households. In Rome itself Greek became the second language of the educated elite. In the days of the late Roman Republic most educated Romans studied Greek. Young wealthy Roman boys were often taught by Greek slaves and sometimes sent to Athens for advanced training.
Plutarch reports that Caesar, when crossing the Rubicon, quoted the Athenian playwright Menander in Greek, saying "let the dice be tossed". Suetonius reports that others have said Caesar's last words were the Greek phrase "καὶ σύ, τέκνον" ("Kai su, teknon?": 'You too, child?').