I guess a good start would be for Gaius Marius to be killed by political rivals before enacting his reforms of the military. This leads to an extended war against the Cimbri and Teutons that further drains Roman manpower and lasts longer and is more destructive.
Couple that with much of the same from that point up until the 60s and 50s BC when the amount of jobless plebs and the lack of reformers in the Senate, with Caesar never rising to the forefront, leads eventually to some form of military reform in order to stave off social rebellion, but no land reform, and eventually, an overwhelming amount of slaves in Italy, along with an overwhelming amount of landless, jobless, plebs.
Eventually, a social rebellion, likely in the 30s BC, along with multiple slave revolts, brings down the Senate. It is replaced with a bunch of warring factions claiming Dictatorial Titles led by ambitious nobles who sided with the rebels. This, combined with marauding bands of slave rebels throughout Italy, leads to sever depopulation and outright poverty as nobody can work the land without being swept up by war. Roman provincial authority slowly collapses abroad into nothingness, and seeing as there was no conquest of Gaul, and no conquest of much of the Near East by Pompey, who was killed in the heavy Roman defeat to Mithridates of Pontus, who crushed Roman Asia, the Republic falls into anarchy at home and withers away abroad. Further Germanic incursions into Italy occur while the warring factions continue to claw at each other over the remnants of the Republic in Italy.
The remnants of the Roman upper class reestablish themselves in Africa and start anew there with a new republic, albeit a very weak one.
EDIT: I didn't see the after Augustus part. My bad.