I too like to have due dates acknowledged and see in my lists which tasks have been completed and which are still left to do.
I quickly worked out that this was not going to be possible in Trello. So I have created a Completed Tasks list, and when I transfer a completed task to this list I change the due date to the date I completed the task. Not ideal, but it does help me with tracking how many tasks I get through each day. Because I also use the labels, I can see how many tasks in each group I get through each day.
Maybe in future editions of Trello this element might be improved.