You would have to commence a plenary action for rent, or use and occupancy, along with damage to the apartment, on a contract or quasi-contract theory. Depending on the amount, you would sue in small claims or in civil court or in Supreme Court (you could check www.nycourts.gov for the appropriate court in your area to commence this case as well as the monetary limit in each available court). You should consult with an attorney in your area as drafting the complaint and figuring out which claims you have may be a bit complicated. Then again, you may not think the legal fees are worth it if you're not suing for much money - that's really a judgment call on your part.
Dear Poughkeepsie Landlord:
You helped make the mess you are in because you chose to have a tenant without a written lease. New York landlord and tenant law does not truly allow for the building up "on account" of a month to month rent, as the general concept is that the month to month tenancy is over when the tenant does not pay the next rent when due. At that point, with the tenancy over, the landlord may "terminate" the tenancy and wait the time out for an eviction proceeding, or serve a rent demand for the rent due, start a nonpayment proceeding, and hope for a money judgment and a possession judgment followed by an eviction if the tenant does not pay.
Now you are left with only a regular civil lawsuit. You have to prove the reasonable value of the use and occupancy for the premises, the value of the damage done to your home by the tenant, win this lawsuit and find your former tenant to collect the judgment.
You may want to consult with an attorney.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.