4-30-2018 -- UPDATE: this achievement has been extremely glitchy for many people, including myself. A new method to get this may have been confirmed just recently however!
You can see Odin's solution for the method, and the short version is to delete your local console save after doing a week's challenge and exiting the game so it will synchronize fresh next time.
Personally this took me 12 weeks to unlock. I had many times where my save data wouldn't appear correctly.
You need 4 weeks in a row of doing a weekly challenge each week. What exactly counts as a 'month' isn't clear, but it appears to be 4 weeks in a row, not a specific month.
Weekly challenges refresh at 7PM PST Saturday, 10PM EST Saturday, or 6AM Sunday GMT.
(That might be one hour later depending on where you live and what Daylight Savings Time is currently)