Call the MOE and ask them for a homeschool information pack. Then answer the questions and submit with a completed application form. Processing time is usually around three weeks, and you can expect that a visitor from the Education Review Office will come out to your house after a couple of months to ask your student a bunch of questions about their education.
If you have any questions, I would recommend you get in contact with the Home Education Foundation (though they're a Christian body) they can still answer any questions you would likely have even about a secular style of education.
Otherwise, I don't see why there would be any problem with continuing your business, but I would suggest consulting a lawyer who specialises in expat tax, so you can figure out if you can avoid taxation for profit you make in both NZ and your home country.
I suggest getting in contact with Aukland Home Educators (http://www.aucklandhomeeducators.org.nz/) - they organise plenty of events where you can meet other homeschool parents and will certainly help with any questions you have.
Good luck with your move! Be sure to explore the Expat Arrivals guide to living and working in New Zealand before you go, and go ahead and post any other questions you may have on this forum.