There are all sorts of back-and-forth issues, but I'll cut to the meat: When you write about an actual person, ALWAYS get permissions from ANY PERSON who will be in your story regarding their materials. And with any adaptation, you NEED permission from the author AND publisher, who often holds the copyright transferred by the author.
Writing a biographical story is challenging from a legal standpoint. While public figures don't have the same degree of privacy afforded common folk, the issues of libel, defamation, profiteering from name or image, and others make this a tricky issue.
If the individual is dead and the people s/he are corresponding with are dead, then it is less problematic. If you are using published letters that are publicly available, such as from an archive, then you only need to respect the estate's copyright. If the letters were published by another author who had to secure permission to publish them, then you are not entitled to quote them.
While documentaries like A&E's Biography, TrueCrimes, etc. weave interesting tales, there are LOTS of permissions and legal work that need to go on behind the scenes. A letter has two parties attached--the sender and recipient. Just because the sender (and/or recipient) is dead, it doesn't mean you don't need clearance from the recipient, their family or vice versa.
Journalists are given slightly greater dispensation but also have lots of legal support. If you are adapting another author's biography, you will definitely want to get his/her permission and the publisher to grant you permission. Some publishers are open to adapting books into screenplays. Others have their own media contacts whom they work with.
I don't mean to make this sound like a project that shouldn't be undertaken. Just put yourself in the place of being the (grand)son/daughter/correspondent of the person being portrayed. (1) What light does it put on them? (2) Would I want people knowing that about me/us? and (3) What's in it for me? The latter is especially true of the author and publisher whose book you're drawing upon. Telling the truth is often more costly then weaving a fantasy. Good luck.
