Sole Roman Emperor
For political and personal reasons, Octavian chose to emphasize his relationship with Julius Caesar by styling himself simply "Imperator Caesar" (whereto the Roman Senate added the honorific Augustus, "Majestic" or "Venerable", in 27 BC), without any of the other elements of his full name. His successor as emperor, his stepson Tiberius, also bore the name as a matter of course; born Tiberius Claudius Nero, he was adopted by Caesar Augustus on 26 June 4 AD, as "Tiberius Julius Caesar". The precedent was set: the Emperor designated his successor by adopting him and giving him the name "Caesar".
The fourth Emperor, Claudius, was the first to assume the name "Caesar" upon accession, without having been adopted by the previous emperor; however, he was at least a member by blood of the Julio-Claudian dynasty, being the maternal great-nephew of Augustus on his mother's side, the nephew of Tiberius, and the uncle of Caligula. Claudius in turn adopted his stepson and grand-nephew Lucius Domitius Ahenobarbus, giving him the name "Caesar" in the traditional way; his stepson would rule as the Emperor Nero. The first emperor to assume the position and the name simultaneously without any real claim to either was the usurper Servius Sulpicius Galba, who took the imperial throne under the name "Servius Galba Imperator Caesar" following the death of the last of the Julio-Claudians, Nero, in 68. Galba helped solidify "Caesar" as the title of the designated heir by giving it to his own adopted heir, Lucius Calpurnius Piso Frugi Licinianus.
Galba's reign did not last long and he was soon deposed by Marcus Otho. Otho did not at first use the title "Caesar" and occasionally used the title "Nero" as emperor, but later adopted the title "Caesar" as well. Otho was then defeated by Aulus Vitellius, who acceded with the name "Aulus Vitellius Germanicus Imperator Augustus". Vitellius did not adopt the cognomen "Caesar" as part of his name and may have intended to replace it with "Germanicus" (he bestowed the name "Germanicus" upon his own son that year).
Nevertheless, Caesar had become such an integral part of the imperial dignity that its place was immediately restored by Titus Flavius Vespasianus ("Vespasian"), whose defeat of Vitellius in 69 put an end to the period of instability and began the Flavian dynasty. Vespasian's son, Titus Flavius Vespasianus became "Titus Flavius Caesar Vespasianus".