I am a NZ Citizen. Do I need a visa to live and run a small business in Melbourne.
I will be traveling back and forth to New Zealand and intending to file my taxes in NZ.
No - you do not need to apply for any visa, so long as you qualify for an SCV (the typical reason for exclusion being if you have a criminal record). See https://movetoaustralia.net/australian-visas for details.
You'll need to be aware of your tax residency status as it will depend on how long you spend in Australia vs how long you spend in NZ as to whether you file tax returns in Australia or NZ.
For the Australian side, see https://www.ato.gov.au/individuals-and-families/coming-to-australia-or-going-overseas/your-tax-residency