After Caligula was killed, Caligula's uncle Claudius took over. He did better than people expected. In his reign the Romans succeeded in conquering England and making it into another province. But Claudius' wife Agrippina poisoned him (according to the Roman historian Suetonius) with bad mushrooms, and he died.
Agrippina the Younger
In 54 AD, therefore, Claudius' stepson Nero took the throne. Nero was only 16, and his mother Agrippina really controlled politics through him (because women could not be tribunes or senators), until Nero was in his mid-20s. But then he decided he would rather rule on his own, and had his mother killed.
Nero may be best known for how he handled the Great Fire at Rome in 64 AD. People were blaming him for the fire, and so he rounded up a lot of Christians and had them burned alive as if the fire was their fault. St. Peter and St. Paul probably also were martyred (killed) during Nero's reign.
After he killed Agrippina, Nero became unpopular with the Senate, and in 68 AD the governor of Spain, Galba, revolted against him and marched his army toward Rome. When it was clear that he was going to lose, Nero killed himself. He was the last ruler from the family of Julius Caesar and Augustus.
For more on the Julio-Claudians:
Classical Rome, by John Clare (1993). For kids, the whole political history from beginning to end.
Oxford First Ancient History, by Roy Burrell (reissued 1997). Easy reading. It skips around a lot, not trying to tell everything, just highlights.
The Romans: From Village to Empire, by Mary Boatwright, Daniel Gargola, and Richard Talbert (2004). Okay, it's a little dry, but it is up to date and has all the facts you could want.
The Roman Revolution, by Ronald Syme (1960). Still a classic.
the Gracchi to Nero: A History of Rome from 133 B.C. to A.D. 68
by H. H. Scullard (1959, 5th edition 1990). Another classic.