Roman imperial history just doesn't divide well into dynasties, frankly. The problems are:
- There are a lot of periods where there was no real dynasty (or at least no stable dynasty) in place (69, 193, 238-285, 455-480)
- Dynasties tended to be extremely short-lived. Yes, there were a number of short-lived Chinese dynasties (the Qin, Xin, and Shun), but they were the exception, not the rule. The average reign of a Chinese dynasty is around 200 years, by my count. Taking the six commonly accepted Roman dynasties (Julio-Claudian, Flavian, Nervan-Antonine, Severan, Constantinian/Neo-Flavian, and Valentinian), you've only got an average reign of 61.5 years. Not a single one of those dynasties remained in power for more than a century.
- Roman historians tended to adhere to the fiction that they were not living in a monarchy until long after it had ceased to be true. The early emperors cultivated the idea that the essential functions of state were still republican, and (probably more importantly) dates were still reckoned by consular office, not regnal years. This structuring of Roman history is what was handed down to us.
- As Dathi THorfinnsson suggests, it's not an especially useful shorthand for understanding trends in Roman history. Chinese dynasties tended to change in response to social pressures, so it's convenient to talk about, say, the Yuan Dynasty as the period of Mongol ascendancy. This doesn't work that well for Rome: there's no obvious distinction between Julio-Claudian Rome and Flavian Rome, or Antonine Rome. Only the Neo-Flavians really straddle a potentially significant change in Roman society, but that distinction is probably better described by talking about the Principate versus the Dominate, or even pagan versus Christian Rome.
Roman dynasties tended to be rather shaky affairs in part because there was no universally accepted principle of hereditary succession. In times of peace of stability, sure, it was expected that an emperor's son (biological or adoptive) would inherit the purple, but I wouldn't say that this was a given. It was, in a sense, an offshoot of the ideas of
auctoritas and patronage: being related to the previous leader gave you the benefit of the doubt under the presumption that you'd inherited some of the power, prestige, and influence of your predecessor, but beyond that you didn't really have an innate claim on power. If someone proved to be a particularly lousy leader, then there was nothing unusual about him being challenged or overthrown. If you want to get at something approximating the Chinese system, you need to establish a more explicitly monarchical system, where the emperor rules because he's the emperor, not just because he's seen (rightly or wrongly) as the most powerful or capable man in the empire.
To get this, you need to ensure a more dramatic break with the republican past. Which isn't easy, since such a huge part of Roman identity was tied up in republicanism (or the illusion of it, at least). Trying to declare yourself king was a quick way to get yourself assassinated: some of the most despised or controversial people in Roman history were accused, rightly or wrongly, or harboring royal ambitions (Appius Cladius Crassus, Tiberius Sempronius Gracchus, etc.). Fears that Julius Caesar was going to declare himself king played a large part in his death, and Augustus only succeeded in instituting a de facto monarchy because he went out of his way to pretend that he wasn't doing so. Maybe Rome gets defeated or even briefly conquered by a powerful monarchical state during the first or second centuries BC, or a much worse Roman revolution leads to a more thorough and complete discrediting of republican government. But, in that case, while you might get a more openly autocratic and monarchical Rome that you could look at in terms of dynastic periods, it would look quite a bit different from OTL Rome, and I can't imagine there'd be that many familiar faces or families.
