There was a play a few months ago apparently just like the one you describe, called "Nevermore" at the Steve Allen Theater. Poe, who died in 1849, has a body of poetry which is now in the public domain, so no rights have to be licensed from Poe or his heirs.
If your projects are about Poe or someone who wrote and died as long ago as his did, both of your projects, a fictionalization of the poet's life, and the recitation of his poetry, don't require acquisition of rights.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
There have been many historical plays such as this, and usually they can be performed without violating intellectual property rights. But you need to be cautious because there still can be problems. Most importantly, in many states there is a "right of publicity" which prohibits exploiting the name or image of a famous person for commercial purposes without obtaining a license from such person or his heirs or representatives. One of the more controversial issues in recent years has been whether the right of publicity continues after death of the famous person. State laws differ on this. In New York, the right of publicity does not continue after death. In California, on the other hand, recent action of the State legislature provides that the right of publicity continues after death. (This became a contentious issue recently in a case involving the estate of Marilyn Monroe which sought to block a photographer from exploiting pictures he had taken of Marilyn Monroe; the outcome of the case turned on whether she was a resident of New York or Califorina---her estate tried to allege that she was a resident of California but when it came time to pay her taxes, she claimed to be a resident of the State of New York, and the New York Federal Court rejected the claim of her Estate that she had been a resident of California). In a state where the right of publicity continues following death of the famous person, the estate of the famous person may continue to attempt to enforce the right of publicity. Thus, your ability to exploit the work of deceased famous persons with impunity may depend on the state in which you intend to perform the play.
Also, you need to be careful when you assert that works are in the public domain. While the writings of Edgar Allen Poe are likely in the public domain, others may have made derivative works based on those writings (plays, books, film, poetry, etc.) that are still protected by copyright. If you borrow heavily from one of these derivative works (still protected by copyright) rather than the original public domain works, you can still violate copyrights held by others. There have been many derivative works based on the original writings of Edgar Allen Poe, and extensive copying from such derivative works can land you in hot water.
The law at the moment is that the state where the "famous" poet was domiciled at the time of death provides the rules as to whether the poet's right of publicity continues or not. If in California, the poet's right of publicity ends 70 years after his or her death. If in NY, the poet has no right of publicity after death [with exceptions]. Other states have their own rules.
As for the copyright in the poems, all that were first published before 1923 are in the public domain and for any first published afterwards a complicated set of rules needs to be applied to determine if they've entered the public domain.
I will simply echo what the previous attorneys have said -- the short answer is that you can use the public domain material in any way you wish.
That said, to the extent your first project has a character playing a famous poet "speaking words written by an author who has imagined the poet's secret life" then the question turns on whether this author's work (i.e., the author imagining the life of the poet) is in the public domain, or whether that material is still subject to copyright protection. If it's still copyright protected, and you are planning on using more than what is colloquially known as the fair-use-snippet, then you may have to come to some sort of licensing agreement. The more you use of this author's protected work, the more likely it is that you will get a call from his lawyer if you don't bring it up preemptively.