Inside the unpaid bills put a-z alphabetical dividers so that the bills can be filed in alphabetical order which makes them easy to find, especially if you have a high volume of bills.
In the paid bills file put Jan-Dec monthly dividers.
When the bills are paid, pull them out of the unpaid bills, keep them in alphabetical order but this time file them into the relevant month in the paid bills file.
What do I mean by relevant month? If you entered a bill into your bookkeeping program at the date that is on the bill then file it into the paid bills file at month/date of the bill.
This all depends on whether or not your bookkeeping program has a creditors/bills option.
If you are only keeping a cash book, then your bill is entered into the bookkeeping program at the date you paid the bill, so you will file it into the month of the payment.
Another method is to simply file all the bills away into one file, whether paid or unpaid, and keep track of the ones you haven’t paid using a spreadsheet.