If the person that you are writing a biography for is already dead, you do not need to ask permission anymore before you start writing the biography. Just make sure that you will gain as many details as possible, and you would like to stick to the facts.
People will not like it if you add some details that are not exactly true. If in case you would like to write the biography of a person who is still alive, you may choose to have an authorized or unauthorized biography. Authorized means that you will be writing with the cooperation of the person. Unauthorized means that you will be writing without the help of the person.