If you're asking "when and why did he also have the name Octavianus?", your friend and Draconis gave you the answer.
If instead you're asking "in what contexts was he called Octavianus, once he gained this name?" the answer is that probably only his enemies, or at least those who weren't exactly allies, called him like that: it's safe to say that he didn't want his modest origins to be highlighted, while his enemies obviously did (notably Mark Antony*). Besides, among the tria nomina the informal one is the praenomen - so after he became Imperator, and later Augustus, virtually no one called him Octavianus. His close relatives (well, at least his wife Livia) may still have called him Gaius, close friends either Gaius or Caesar. After becoming Augustus, everyone else called him Divus Augustus.
Going on a tangent, it is interesting to note that by the start of the 2nd century AD, the informal name had become the cognomen, due to the extreme scarcity of praenomina faced with the population explosion, which had made them quite impractical.
*EDIT: As a matter of fact, Suetonius tells us that Antony went even further, calling him Thurinus with disdain - this was his original cognomen, though not even Suetonius is sure about the origin of it.
It was instead Cicero, and presumably other similar republican figures, who called him Octavianus - it makes sense for him to refer to Caesar's adoption, because Cicero hoped young Octavius could be manipulated to restore the Republic, and had a great respect for Caesar.