Why did Nero kill his own mother? Was it true that she was dangerous to the point where it was "her life or his?" Did he have "good" reasons (in the sense of being understandable or acceptable at the time) for the killing of other people, and a persecution of the Christians.
Nero's mother always wanted to get power, note that she was married to emperor Claudius, he already had a son, however, Agripina convinced Claudius to actually allow her son (Nero) to be the emperor.
After that Claudius died in obscure circumstances.
At that time Nero was really a kid, so Agripina actually had a lot of power, as Nero became an adult, Agripina had to effectively let that power go, but I guess that is difficult for someone used to always use and enjoy that power.
If in top of that you add the character of Nero, you have the perfect cocktail.