There is a simple and vexing problem with any discussion of the Roman emperors: the sources. While archaeology and the surviving written sources create a reasonably clear basis for understanding the major political events of the Julian dynasty, the biographical details are much more difficult. All of the surviving written accounts about the lives of the Julian emperors were written many decades, in some cases more than a century, after their reign. In turn, the two most important biographers, Tacitus and Suetonius, detested the actions and the character of the Julians, and thus their accounts are rife with scandalous anecdotes that may or may not have any basis in historical truth (Tacitus is universally regarded as the more reliable, although Suetonius’s The Twelve Caesars does make for very entertaining reading). Thus, the biographical sketches below are an attempt to summarize what is known for sure, along with some notes on the scandalous assertions that may be at least partly fabricated.
When Augustus died in 14 CE, his stepson Tiberius (r. 14 – 37 CE) became emperor. While it was possible that the senate might have tried to reassert its power, there was no political will to do so. Only idealistic or embittered senators really dreamed of restoring the Republic, and a coup would have been rejected by the vast majority of Roman citizens. Under the Caesars, after all, the empire had never been more powerful or wealthy. Genuine concessions had been made to the common people, especially soldiers, and the only people who really lost out in the short term were the old elite families of patricians, who no longer had political power independent of the emperor (although they certainly retained their wealth and status).
Tiberius began his rule as a cautious leader who put on a show of only reluctantly following in Augustus's footsteps as emperor. He was a reasonably competent emperor for over a decade, delegating decisions to the senate and ensuring that the empire remained secure and financially solvent. In addition, he oversaw a momentous change to the priorities of the Roman state: the Roman Empire no longer embarked on a sustained campaign of expansion as it had ever since the early decades of the Republic half a millennium earlier. This does not appear to have been a conscious policy choice on the part of Tiberius, but instead a shift in priorities: the senate was now staffed by land-owning elites who did not predicate their identities on warfare, and Tiberius himself saw little benefit in warring against Persia or invading Germany (he also feared that successful generals might threaten his power, at one point ordering one to call off a war in Germany). The Empire would continue to expand at times in the following centuries, but never to the degree or at the pace that it had under the Republic.
Eventually, Tiberius retreated to a private estate on the island of Capri (off of the west coast of Italy). Suetonius’s biography would have it that on Capri, Tiberius indulged his penchant for bloodshed and sexual abuse, which is highly questionable - what is not questionable is that Tiberius became embittered and suspicious, ordering the murders of various would-be claimants to his throne back in Rome, and sometimes ignoring affairs of state. When he died, much to the relief of the Roman populace, great hopes were pinned on his heir.
That heir was Gaius (r. 37 - 41 CE), much better known as "Caligula," literally meaning “little boots” but which translates best as "bootsie." As a boy, Caligula moved with his father, a famous and well-liked general related by marriage to the Julians, from army camp to army camp. While he did so he liked to dress up in miniature legionnaire combat boots; hence, he was affectionately dubbed "Bootsie" by the troops (one notable translation of the work of Suetonius by Robert Graves translates Caligula as "Bootikins" instead).
Even if some of the stories of his personal sadism are exaggerated, there is no doubt that Caligula was a disastrous emperor. According to the biographers, Caligula quickly earned a reputation for cruelty and megalomania, enjoying executions (or simple murders) as forms of entertainment and spending vast sums on shows of power. Convinced of his own godhood, Caligula had the heads of statues of the gods removed and replaced with his own head. He liked to appear in public dressed as various gods or goddesses; one of his high priests was his horse, Incitatus, whom he supposedly appointed as a Roman consul. He staged an invasion of northern Gaul of no tactical significance which culminated in a Triumph (military parade, traditionally one of the greatest demonstrations of power and glory of a victorious general) back in Rome.
Much of the scandalous gossip about him, historically, is because he was unquestionably the enemy of the senate, seeing potential traitors everywhere and inflicting waves of executions against former supporters. He used trials for treason to enrich himself after squandering the treasury on buildings and public games. He also made senators wait on him dressed as slaves, and demanded that he be addressed as “dominus et deus,” meaning "master and god." He was finally murdered by a group of senators and guardsmen.
The next emperor was Claudius (r. 41 – 54 CE), the one unquestionably competent emperor of the Julian line after Augustus. Claudius had survived palace intrigues because he walked with a limp and spoke with a pronounced stutter; he was widely considered to be a simpleton, whereas he was actually highly intelligent. Once in power Claudius proved himself a competent and refreshingly sane emperor, ending the waves of terror Caligula had unleashed. He went on to oversee the conquest of England, first begun by Julius Caesar decades earlier. He was also a scholar, mastering the Etruscan and Punic languages and writing histories of those two civilizations (now lost, unfortunately). He restored the imperial treasury, depleted by Tiberius and Caligula, and maintained the Roman borders. He also established a true bureaucracy to manage the vast empire and began the process of formally distinguishing between the personal wealth of the emperor and the official budget of the Roman state.
According to Roman historians, Claudius was eventually betrayed and poisoned by his wife, who sought to have her son from another marriage become emperor. That son was Nero. Nero (r. 54 – 68 CE) was another Julian who acquired a terrible historical reputation; while he was fairly popular during his first few years as emperor, he eventually succumbed to a Caligula-like tendency of having elite Romans (including his domineering mother) killed. In 64 CE, a huge fire nearly destroyed the city, which was largely built out of wood. This led to the legend of Nero "playing his fiddle while Rome burned" - in fact, in the fire's aftermath Nero had shelters built for the homeless and set about rebuilding the roughly half of the city that had been destroyed, using concrete buildings and grid-based streets. That said, he did use space cleared by the fire to begin the construction of a gigantic new palace in the middle of Rome called the "golden house," into which he poured state revenues.
Nero’s terrible reputation arose from the fact that he unquestionably hounded and persecuted elite Romans, using a law called the Maiestas that made it illegal to slander the emperor to extract huge amounts of money from senators and equestrians. He also ordered imagined rivals and former advisors to kill themselves, probably out of mere jealousy. Besides Roman elites, his other major target was the early Christian movement, whom he blamed for the fire in Rome and relentlessly persecuted (thousands were killed in the gladiatorial arena, ripped apart by wild animals). Thus, the two groups in the position to write Nero's history - elite Romans and early Christians - had every reason to hate him. In addition, Nero took great pride in being an actor and musician, two professions that were considered by Roman elites to be akin to prostitution. His artistic indulgences were thus scandalous violations of elite sensibilities. After completely losing the support of both the army and the senate, Nero committed suicide in 68 CE.
Another note on the sources: what the "bad" emperors of the Julian line (Tiberius, Caligula, and Nero) had in common is that they violated the old traditions of Romanitas, squandering wealth and glorifying themselves in various ways, thus inspiring hostility from many elite Romans, the senate included. Since it was other elite Romans (albeit many years later) who became their biographers, we in the present cannot help but have a skewed view of their conduct. Historians have rehabilitated much of the rule of Tiberius and (to a lesser extent) Nero in particular, arguing that even if they were at loggerheads with the senate at various times and probably did unfairly prosecute at least some senators, they did a decent job of running the empire as well.