To change the name on your birth certificate you will need to fill out a name change statutory declaration form.
This declaration needs to be signed in front of a Justice of the Peace or other authorised person.
The information required includes your new names, and your birth names. No other documentation is required for this unless you have previously changed your name, or if your citizenship is in another country. In those cases, some additional info may be required.
This form also contains an application for a copy of your birth certificate with your new legal name.
You will need to fill in this form and send it in the post to the address given on the form, to receive your new birth certificate. The fee for the name change and a copy of your new birth certificate is around $150.
Department of Internal Affairs have the answers to frequently asked questions regarding name changes on there website.
Here is an example of a statutory declaration, and how to fill one in.