hi YannisHD
You've already had some great advice from AnTyx & I assume you've already browsed this thread (if not, I suggest you do as it will still be relevant - NZ doesn't change that much).
Allow for being jet-lagged & drive left - care is needed as NZ drivers aren't the best.
NZ doesn't have big highways & the national speed limit is 60mph. The police love tourist roads and unmarked vans have plate recognition. As a tourist, you are not exempt from a hefty fine.
Your route will take you through towns with limits of 30-40mph and you'll probably stop a lot for great coffee, good food and to capture that view (so many views).
North Island - warmer/humid the further north you go. Rolling green hills with the occasional volcano. Geothermal activity around Rotorua and Taupo. More people = busy roads.
South Island - drier climate with the exception of the west coast. Home to the Southern Alps where the highest peaks are covered in snow all-year. Amazing roads for bikers. Less people = less traffic. No motorways, twisty country roads where the pace is slow.
Fly to Christchurch, Queenstown or Dunedin then hire a bike. Plan short distances. Bring your wet riding gear & be grateful if you don't need it (this applies to any month of the year).
If you really want to see some of the NI, to reduce costs, you could hire a car. It's much cheaper & costs can be shared. Many hire companies would be happy for you to drive to CHC or ZQN and leave it there. The ferry cost will be similar to 2 bikes.
If flying, book early. There are only 2 airlines so use WebJet to browse for deals.
If you use Facebook, there's plenty of bike sites where advice and friends can be found