Why did emperor Nero behave so badly?

We can only speculate.
He grew up in an intellectually gifted family - his ancestors Julius Caesar, Augustus, Tiberius, Caligula and Claudius were all brilliant and very privileged. Growing up in a bubble of aristocratic privilege tends to make a person narcissistic. Evidence that he was can be found in his love of acting and performing on stage before the general public, a terribly vulgar thing for an Emperor to do. But he lived for applause and attention. In addition Roman politics was brutal, to say the least. He knew he had to escape the influence of his domineering mother, and decided to have her murdered. Without her influence, he went off the deep end and governed as if he were a celebrated actor playing the part of Emperor. This is a seriously distorted way to live.
His persecution of the Christians is not what shocked his contemporaries. Nero simply didn't recognize limits of any kind, and this gave him ample scope for his sadism.