He was preparing an invasion of Parthia, so assuming that goes ahead, the most likely result is that he gets defeated (as had happened to Crassus before hand and happened to Antony afterwards), which robs him of a lot of legitimacy and likely leaves him in an untenable position in Rome, which might force him into exile (probably to Egypt, where a new Julian dynasty rules as Pharaos with Ceasarion succeeding Ceaser and Cleopatra). Antony probably joins him in Egypt, and with their combined talent, they could make Egypt into a major power in the eastern med, powerful enough to keep Rome away (that or bribe Rome with free grain). ITTL, Octavian doesn't have his meteoric rise to power, but he's just too cunning to not get involved in politics, although he might end up as Cicero's protegé in the Senate, probably leading a distinguished senatorial career.
After Ceaser and Pompey though, I can't really think of any general capable of challenging the senate, so that might buy the republic enough time to make it all the way to the CE.
If he wins in Parthia, his prestige and wealth will be at a level where moving against him is impossible for the senate, and he rules Rome with an iron fist until he dies. After that though, there's probably going to be a bloody civil war much in the same vein as OTL, with Antony, Ceasarion and Octavian jockeying for position. Octavian is definitely in the weakest position in this case, and he'll probably try to get into Ceasarion's good graces and tell him that Antony is the greater threat to his claims than he is (which would be true, since Antony would have Ceaser's army, and Octavian lacks the recognition being Ceaser's heir granted him). If they finish off Antony, I can see Octavian subsequently forging something that is more or less like the principate he created in OTL but with Ceasarion as the titular "monarch" (I doubt he'd actually call himself king of emperor, just like Octavian did), with Octavian as his Prime Minister, and likely heir if ever Ceasarion dies without issue.
Octavian would probably try and maintain at least some of the senate's powers, and as a result maybe the consuls become co-prime ministers for the emperor Ceasarion (maybe Agrippa gest the other consulship, thus ensuring it remains in the hands of the Julian faction).