This achievement requires you to talk to Calendar Man on certain dates. You can find Calendar Man in the basement of the court house (It's in the back of the building where you rescue Cat Woman in the beginning of the game).
The dates (for the year 2012) are:
May 13 - (2013 = May 12)
June 17 - (2013 = June 16)
August 16 - (2013 = no idea if the date is the same, Feast of St. Roch)
September 3 - (2013 = September 2)
November 22 - (2013 = November 28)
You will have to wait for these dates, or go offline and set the date of your xbox manually (See Lainey80's comment for detailed description). Everytime you talked with him on a right date, the appropriate month will be marked on the calendar next to the cell.
- It appears that sometimes different dates are needed due to the year settings of the console, just check the calendar after you did a date that is the same every year (e.g. X-mas day: December 25) and all the other dates that are needed should be marked on the calendar (Credits go to stevivor and Shinwasha for finding this out) -