What does it actually mean to "break bad"?

Breaking bad is a slang phrase meaning to defy authority or to challenge conventions.

Source Urban Dictionary

I have just started watching initial 4-5 episodes and story starts with protagonist starting to work illegally.

Its basically about chemistry teacher who is very simple guy but he is came to know that he has cancer. He is only support for his family and he also have one son who is young and differently able.

So, this fellow comes across one of his old student and starts creating Meth.

You can say its his story of defing authority or law, in other words, Watler is breaking bad.

Lead actor Bryan Cranston stated in an interview that: "The term 'breaking bad' is a southern colloquialism and it means when someone who has taken a turn off the path of the straight and narrow, when they've gone wrong. And that could be for that day or for a lifetime."

_{[Source: http://breakingbad.wikia.com/wiki/Breaking_Bad]}

Showcreaters intially thought that this is very common term as mentioned in this article:-

Show creator Vince Gilligan has said that he had thought it was a commonly used phrase when he decided to use it as a title, not knowing that the expression was a Southern regionalism from the area in Virginia from which he hails. It means “to raise hell,” he says, as in “I was out the other night at the bar…and I really broke bad.”