Julius and Claudius were two Roman family names; in classical Latin, they came second. Roman family names were inherited from father to son, but a Roman aristocrat could – either during his life or in his will – adopt an heir if he lacked a natural son. In accordance with Roman naming conventions, the adopted son would replace his original family name with the name of his adopted family. A famous example of this custom is Julius Caesar's adoption of his great-nephew, Gaius Octavius.
Augustus (Imperator Caesar Divi Filius Augustus), as Caesar's adopted son and heir, discarded the family name of his natural father and initially renamed himself "Gaius Julius Caesar" after his adoptive father. It was also customary for the adopted son to acknowledge his original family by adding an extra name at the end of his new name. As such, Augustus' adopted name would have been "Gaius Julius Caesar Octavianus". However, there is no evidence that he ever used the name Octavianus.
Following Augustus' ascension as the first emperor of the Roman Empire in 27 BC, his family became a de facto royal house, known in historiography as the "Julio-Claudian dynasty". For various reasons, the Julio-Claudians followed in the example of Julius Caesar and Augustus by utilizing adoption as a tool for dynastic succession. The next four emperors were closely related through a combination of blood relation, marriage and adoption.
Tiberius (Tiberius Caesar Divi Augusti Filius Augustus), a Claudian by birth, became Augustus' stepson after the latter's marriage to Livia, who divorced Tiberius' natural father in the process. Tiberius' connection to the Julian side of the Imperial family grew closer when he married Augustus' only daughter, Julia the Elder. He ultimately succeeded Augustus as emperor in AD 14 after becoming his stepfather's adopted son and heir.
Caligula (Gaius Julius Caesar Augustus Germanicus) was born into the Julian and Claudian branches of the Imperial family, thereby making him the first actual "Julio-Claudian" emperor. His father, Germanicus, was the son of Nero Claudius Drusus and Antonia Minor, the son of Livia and the daughter of Octavia Minor respectively. Germanicus was also a great-nephew of Augustus on his mother's side and nephew of Tiberius on his father's side. His wife, Agrippina the Elder, was a granddaughter of Augustus. Through Agrippina, Germanicus' children – including Caligula – were Augustus' great-grandchildren. When Augustus adopted Tiberius, the latter was required to adopt his brother's eldest son as well, thus allowing Germanicus' side of the Imperial family to inherit the Julius nomen.
Claudius (Tiberius Claudius Caesar Augustus Germanicus), the younger brother of Germanicus, was a Claudian on the side of his father, Nero Claudius Drusus, younger brother of Tiberius. However, he was also related to the Julian branch of the Imperial family through his mother, Antonia Minor. As a son of Antonia, Claudius was a great-nephew of Augustus. Moreover, he was also Augustus' step-grandson due to the fact that his father was a stepson of Augustus. Unlike Tiberius and Germanicus, both of whom were born as Claudians and became adopted Julians, Claudius was not adopted into the Julian family. Upon becoming emperor, however, he added the Julian-affiliated cognomen Caesar to his full name.
Nero (Nero Claudius Caesar Augustus Germanicus) was a great-great-grandson of Augustus and Livia through his mother, Agrippina the Younger. The younger Agrippina was a daughter of Germanicus and Agrippina the Elder, as well as Caligula's sister. Through his mother, Nero was related by blood to the Julian and Claudian branches of the Imperial family. However, he was born into the Domitii Ahenobarbi on his father's side. Nero became a Claudian in name as a result of Agrippina's marriage to her uncle, Claudius, who ultimately adopted her son as his own. He succeeded Claudius in AD 54, becoming the last direct descendant of Augustus to rule the Roman Empire. Within a year of Nero's suicide in AD 68, the Julio-Claudian dynasty was succeeded by the Flavian emperors following a brief civil war over the vacant Imperial throne.