The Republic is dead, one way or another. Without the Caesarean model of Dictatura, I'm not sure whether a stable system can be achieved at all. It took a charismatic warleader to bring Rome into the modern age (if you look at comparable instances, it is almost incredible how long they managed to cling to the impractical polis model - at that point, Greece had long switched to basileia). Without the switch, the Imperium might simply fall apart like the Delian League did.
Of course it would depend on who does it, too. I suspect the most likely culprit would be someone close to him, but jealous of his success. Pompey comes to mind if it happens before their war. Or Sextus Pompeius, after the war, to avenge his father. Alternatively, maybe Lepidus or Marcus Antonius figure they can get away with replacing Caesar. I don't think either had it in them, though if anyone, Marcus Antonius might have stood a chance. If it's Pompey, he might try to rebuild his father's clientela and make himself dictator perpetuus, but the Caesarians would likely prevent that. In that case, a collapse into disorder is likeliest, especially after the star. The people would panic and riot.