Camping is a very popular activity in New Zealand. DOC and private campgrounds are everywhere. There are also a heap of regional parks administered by various regional councils. Check http://www.arc.govt.nz/
for some in the Auckland region as an example.
Unfortunately, New Zealand has fewer and fewer private camping grounds all the time, as they are brought up and turned into luxury housing (especially the beachfront ones). But the ones that remain usually have full kitchen facilities and are generally of a high standard. Watch out for campgrounds in urban centers though - some are what you would call a trailer park.
Stealth camping is also tolerated as long as you are tidy and minimize impact. At worst, you will get frowned upon by a ranger, but the ranger will also be happy that you are warm and well fed rather than hypothermic and the subject of a search and rescue mission.
Don't forget to ask a local or farmer. You will likely get a spot of lawn to camp on, but may also get a full dinner, breakfast and a hot shower if the story of your tour so far is entertaining enough...