Click to open the item from your board.  You'll see a task preview that might look like this (if it's just a task or deal with no email attached the note will be on the left in place of the email).

Click on the middle icon in the right panel below the notes to add a due date.

The date will now be visible on your board.

If the card is overdue, the date will show in red.

You can also view your board by due date using the date button at the top:

Related: Using Date view