From a legal point of view you don't need anyone's permission but unless you're simply going to rehash what other people have written already, or unless you actually knew this person yourself, you're going to have to contact his friends/ family/ fellow actors etc. It would be (at the very least) polite to approch his next of kin first and explain what you would lke to do and why. You may find that they have a similar project in mind in which case you may have to back away. On the other hand you may find that they would love to see their relative 'in print' but haven't the time or the resources to do it. They could well be the very best source of previously unpublished facts/ letters etc without which your book would be very incomplete and with which it might be much more interesting. First step (probably) should be to find out who they are. You're going to find writing the book a bit difficult without mentioning his family!
How long ago did they die? Most obituaries will mention wives, partners, children etc. Did they have a Who's Who entry? What does the internet say?