I was just reading up on Julius caesar and several other early Roman emperors and it seems like they rarely had biological children. Caesar had one daughter and a son by cleopatra, Octavian not had one daughter, and so on. Why is that? You think they would've put more emphasis on continuing the dynasty. Maybe it's just that I've playing to much ck2 and watching too much game of thrones, I don't know but it just seems odd that these emperors didn't really try and have plenty of children to pass the throne onto.
At least as far as the Julio-Claudians, since that sounds like what you're interested in... They did try!
Julius and Augustus both actually promoted "rewards" for people who had lots of children, and Augustus made adultery illegal and heavily punished, in order to try to increase the general birthrate.
All of the Julio-Claudian emperors had at least one child, usually more like three or four. I wouldn't be surprised if there were more stillbirths and miscarriages that weren't recorded; Roman records can be a little dicey.
The "sane" emperors often put lots of emphasis on continuing the dynasty; it was just a brutal and unstable time. And it didn't help that the "crazy" emperors like Tiberius and Caligula tended to murder whoever looked like a threat - family members were at the top of that list.
For instance, Augustus planned to have his grandkids, Gaius and Lucius, co-rule. But they died. He then planned to have his grandson Agrippa Postumus rule. But he was exiled and disowned, possibly for treason. Then he planned to have his step-son Tiberius rule, but he was exiled too, possibly for defying Augustus. Then he planned to have his step-son Drusus rule, but he died. Then he planned to have his grandson Germanicus rule, but he died. Then he was basically stuck with begging Tiberius to come back at the last minute, and that ended badly.
It was also less cemented in the Roman Empire that one's heir must be one's biological issue. The idea that a particular bloodline was divinely ordained didn't exist yet. Emperors tended to favor their own relatives, but what they really emphasized was continuity of governance. They didn't want the country to end up adrift without a captain, so they had to get some heir trained and already co-ruling by the time the current emperor died.
(This is also why you don't see pre-teen Roman emperors.)
I can get into the later empire if you're interested, but basically... The Roman Empire was way more unstable than you're probably imagining. There were small islands of dynastic rule, punctuated by big floes of "whoever has the biggest army now, whoops, it's been six months and now you're dead." And when you're murdered after six months, you don't really have time to father and raise heirs.
ETA: To expand slightly on the last point: The longest dynasty during the (unified) Roman Empire lasted for seven emperors. Compare this to the English Plantagenet dynasty, which had 14 kings and was followed by the semi-related Tudor dynasty. Or the French Capetian dynasty, which had 36.
Rome didn't have the same tradition of "the king is dead, long live the king" that medieval and early modern Europe did; every transition of power was dangerous and uncertain, and there was no guarantee that the next king would be a blood relative.
This previous thread may shed some light here. r/AskHistorians/comments/287h6e/why_did_so_many_roman_emperors_adopt_their/