Countries will extradite their citizens if they've been charged with a crime, but it has to be a crime in both countries, and it needs to be of some severity. Parking tickets aren't enough to get me extradited from Canada, even though not paying them is a misdemanor. Similarly, charging me with blasphemy in Iran and asking for me to extradited won't work either.
NZ needs to have made copyright infringement an indictable offence, and they need to have done so before Mr Dotcom was charged.
If not, and if they wish to get rid of him, they need to ensure somehow that he doesn't have a lawyer, and then hope he can't defend himself adequately.
This is a classic way to get a proponent of X into trouble: get them to say under what circumstances X would be breaking the law, and assert they were a proponent of breaking the law. Another is ordering someone not to do something legal, then charge them with disobedience. A third is to ask them if they had (ever) broken the law, then charge them with lying if they had but the statute of limitations had run out.
All are hard to defend against, as they're constructed half-truths. None addresses the propriety, truth or desirability of the original action, only the consequent, so a court can sometimes be tricked into ruling narrowly on the second part alone.
Life is a healthy respect for mother nature laced with greed.