Ah. This might be a problem. NJ is one of the most tenant-friendly states in the country. A landlord cannot evict a tenant unless they fall into one of 16 specific categories. You can read the list on pages 36 through 43 here:
http://www.nj.gov/dca/divisions/code..._lti/t_i_r.pdf
If the property you are leasing is located in an owner-occupied 2 or 3 family home then that law does not apply and the method of eviction is shown on page 43 at the above link.
If this is not an owner-occupied 2 or 3 family home then to evict for habitual late payment you would have to follow the procedure listed in item J. This method only works if the tenant pays late every month.
Since this is a month-to-month tenancy you would probably have better luck giving the tenant a 30-day written Notice to Quit and Notice of Rent Increase. If the tenant fails to pay the new rent amount on time you can immediately file a lawsuit for eviction without any additional notices. This method is detailed in item F.