This is a great question. Knowing Atticus's personality and character, the letter should be respectful but firm. In other words, it should be direct but in a way that is non-threatening. It will also be culturally sensitive. Atticus knows that he might be dealing with more racist people who do not see eye to eye with him. So, he will ask for information in a way that they can understand. For example, he might ask for details of Tom's death in order to bring closure to the tragedy.
First, he might want to ask for information to bring comfort to Tom's family and friends. For example, Atticus might want to know Tom's last words to comfort Tom's grieving widow, children, and parents. Moreover, the warden, if he is a decent man, would give this information.
Second, Atticus might want to ask for Tom's belongings. Did he leave anything? More importantly, did he write anything to his family? Again, a decent man would not begrudge such things.
In conclusion, a gentle (non-accusatory) yet direct letter would not only fit Atticus's personality but also yield the best results.