A major issue is that Caesar himself was a controversial figure, who had gone against tradition repeatedly, and was suspected of wanting to commit the biggest no-no any Roman could imagine: making himself king. As such, relying on being the son of Caeesar cannot possibly be enough to get the sort of legitimacy you'd need.
The relgious angle might work better. The army was always a bit of a hotbed of cultism, and Marcus Antonius had cred there. If Caesarion can endear himself to the army, and get a religious mystique going that he is a godly-descended saviour figure, this can create something powerful: an army that is both organised and religiously motivated. Historically, that has been a successful combo. The religious approach isn't even that strange: times of trouble often cause religious intensity (and creative cultism). Caesarion could easily use the message that the recent period of chaos has been cause due to Rome's corruption and decadence (the big complaints of the late republic), and that he is there to restore Rome to its virtuous roots. He can cloak his foreign ancestry in the guise of something useful, to which a lot of Romans in the period were open: esoteric eastern knowledge, which will help in restoring the ancient and mythic stature of Rome. (Augustus using the Aeneis 'orgin story', which traced Rome's ancestry to the mythic east, as a way of legitimising his own rule and traditional cred worked in much the same way, really.)
In any event, the initial support base will have to be the army. They're most likely to be receptive, and least likely to outright reject him just because he's part-foreigner. Whether he then broadens his base to the common man or to the elite must depend entirely on which of those groups is more open to supporting him. Considering Caesar's more populist politics, I somewhat expect the 'comman man' to be more eager to flock to his banner. On the other hand, new-fangled cults were more popular with the elite, whereas the lower classes were more 'religiously conservative', so to speak.