Thank you for the question. So the vaunted, and rarely seen, $0 result. I have only seen it once on an assisted phone call in four years, and the customer could not understand how. Eventually after a half hour they understood.
It is highly unlikely that you would get a zero dollar result. Just fill in the W-4 using these two resources and hope that it is $0.
See both:
TurboTax's Tax Caster tax calculator
Turbo tax has a W-4 calculator available in which you can enter income and tax withholding information to calculate what your withholding should be for the year. The more precise your input the better the results with the calculator.
After you get the desired results you may print out a blank W4 form to fill out and give to your employer.