You could indeed spend all of your time on the North Island, and wouldn't necessarily go wrong if you did, but it would be a shame to go all the way to NZ and not see at least some of the wonders of the South Island - especially the area in and around Queenstown with it's easy access to world class hiking, Te Anau, Milford Sound, and numerous LOTR film locations. Aoraki/Mt Cook is worth a visit too - absolutely gorgeous mountain scenery.
Christchurch is still rebuilding from the earthquakes of 2011. The whole city seemed to be one giant construction zone when we visited last year - I'd give it a miss if I were you. Likewise, I wouldn't waste any of your limited time on the Interislander ferry. Better to use that day to explore some of the many beautiful sights the country has to offer.
In April you'll be visiting during the shoulder season when most of the crowds of summer have started to disperse and the skiiers haven't yet descended on the Southern Alps. Would think that you could get pretty good deals on a camper van rental if that's really what you want to do. Traditional accommodations should be pretty cheap too if you choose to forego roughing it.
There are several direct flights per day connecting Auckland with either Christchurch or Queenstown, making for easy connections between the two.
New Zealand is (in our opinion) the most beautiful country on earth. Suggest studying several good guidebooks to see what's on offer and then structure an itinerary according to your particular interests.