If you do not have a written lease and he has been paying month to month, the courts will likely find that you will have an "implied" month to month lease. A written document is not required. Based Florida statutes, when the tenancy is from month to month, to termiate the lease you need to give not less than 15 days’ written notice prior to the end of any monthly period. If he stays past that you will need to follow the procedure for eviction through the sheriffs department. I am not going through that process now. It is important that you do not try to do anything to try to make him leave on your own, such turning off the power or utilities to his room.
As for the threats, report those to law enforcement immediately. If they were verbal threats they still can be found to be extortion which is a felony. Extortion is threatening (verbally or written) to injure someone, their property, or their reputation or threatening to expose secrets or disgrace them in order to gain money or to get the alleged victim to do something. Without knowing details, its hard to give you specific advise regarding these threats, but it is not something to be taken lightly. It is important that you evaluate your safety carefully.
Due the nature of you issue, I would suggest you speak with an attorney to review the specifics of you situation.
As regards the sex offender tenant:
If the tenant pays on the first of the month, then you have to send the termination letter via certified mail and he has to receive it more than 15 days prior to the beginning of the next monthly term. So, for the end of December, you would have to make sure he received it by (let's just say the 15th to make sure you really make the deadline.) Mail it at least by December 10th to give the letter carrier a few chances to catch this tenant.
You can give a Pay or Quit Notice ( aka three day notice) but make SURE to properly fill it out. Don't just use the clerk of court's handout, as it is missing a statutory element and the judge will see that. Once you serve this, the tenant has time to pay and even if you file, he can still deposit the money into the court registry and he gets to continue living there.
If the tenant is in violation of the Florida Statutes Chapter 83 for residential tenancy, then you can give him a 7 day Cure or Lease will be Terminated Letter.
I would recommend properly executing and sending a Pay or Quit Notice immediately and also mailing cert. mail a Termination Notice. That way, you are covering your bases two ways.
As to the tenant threatening you, do as the first answering attorney suggests and call 911.
First, how did you find out this person is a registered sex offender? Make sure you are correct in this identification. In Florida, registered sex offenders are required to register their place of residence with the local Sheriff's Office. If this person is in fact a registered sex offender, and has been living in this motel without registering this place as his residence, it's possible that he has violated this law. In addition, if someone has made a threat of violence against you, you should call the police. Be safe and protect yourself!
My answers on Avvo.com are intended to help clarify the legal process in your situation to the extent possible on this website. Criminal law can be very complicated and very serious. Every case is different and my answers are not intended to serve as legal advice unless so stated. If you have been charged with a crime you could be placed on probation or face incarceration. Please contact an attorney if you think you need legal advice in a criminal matter.