I would not rely on either non-payment or accumulation of trash as grounds for eviction. Rather, I would proceed to terminate the tenancy at will and then follow that up with a complaint for eviction. If the tenant pays her rent (or is supposed to pay her rent) monthly and on the first of the month, you can terminate the tenancy as of June 30 if you have her served with a notice of termination on or before May 31. But the notice needs to have certain magical language and is best served by a constable.
Just to round out the analysis, you don't want to use trash as a basis for eviction because (a) she might "see the light" and remedy the problem as soon as she receives notice and (b) judges see trash-related reasons for eviction as generally being flimsy.
I agree with the above answers, but would like to note that in order to terminate a tenancy at will, there are different procedures than a tenancy under a lease.
First, in order to terminate a tenancy at will you must terminate with one rental period's notice and it must end on a rent date.
Additionally, the landlord must prove the receipt of the notice to quit. In other words, send it by certified mail, or with a constable or sheriff.
Then after the notice to quit has expired, you may get the required form for Summary Process and Complaint from either the Housing Court or District Court and fill it out. You may then serve the tenant, in hand, leave it at their home, or mail it through the constable/Sherriff. You then have to file the complaint with the Court and pay the filing fee. You may then use the court system to evict your tenant.
The Court generally makes these steps easy enough to be done without a lawyer, however having a lawyer certainly makes the process easier, smother, and ensures you do not miss anything.
Best of luck