You could set-up a button/script that would search for all bill records with a due date of today or earlier.
Of course, you might want to have it search for due dates earlier than a week from today, so you also see those that are about to be due. You could get 'fancier' by showing the future, current and past due records in different colors or in different sections of a summarized sorted list.
You might build a relationship that shows a list of due bills on a 'dashboard' or main screen of your system. This would probably be based on a global field that every day is set to whatever date you want to see bills on or before. The portal's relationship would include only unpaid bills, perhaps using a flag field as the 2nd predicate in that portal's relationship.
Are you thinking automated reminders? Maybe a server-side script to find the due bills, then send an email to you that lists them.