Buying a car in NewZealand is quite easy. Here is what you have to do:

1. Find a car, make sure it has a current WoF and Registration (and a RUC-Licence for diesel-vehicles).
The old owner of the car has a Certificate of Ownership. You fill in your details as the new owner, then you both have to sign it. Since most backpackers don't have a permanent address choose one where you will be in about 2-3 weeks time (e.g. a hostel). It is used to send the new Certificate of Ownership to you. The upper part of the paper is for you to take to the post office, the lower part needs to be put in a mailbox. You can check if a car is stolen or has money owing by calling AutoCheck on 0800 843 848 (cost: $25, payment by creditcard). You'll need the Licence-plate number of the car for this service.

2. Change ownership at any post-office
You will need the upper part of the paper and some form of ID, and a new Certificate of ownership will be sent to your postal address.

3. If you want, get insurance for your car
Your Registration already covers you for damage to people, but not for damage to property (e.g damage to cars). There are several insurance-companies, usually you will have to buy a full years cover but some give you a refund if you cancel earlier.

Things to do while you have your car:

Your must have Registration and a current WoF for your car at all times. If your WOF expires go to any garage or testing station, they will check your car and give you a new one. If your Registration expires go to any post office and renew it. (To renew your Registration you must have a current WoF). For diesel-cars you also have to pay road-user-charges (RUC). You pay per 1000km also at the post office.

What it will cost to have a car:

Change of ownership: $ 9.20
WOF: around $ 30 for half a year (varies a little depending on where you have it done)
Registration: (depends on type of car and periode)

3 month
6 month
12 month

Road user charges-RUC (diesel-cars only): about $30 - $35 per 1000km (depends on your car and how many km you buy)
Insurance: depends on insurance

Links for more information:

Land Transport and Safety Authority
Automobile Accociation AA