Talk to Calendar Man (in the basement of the Solomon Wayne courthouse, i.e. the Two-Face/Catwoman location at the beginning) on the dates noted below.
(list thanks to NL WHooPwhOOp)
The dates (for the year 2012) are:
Boosting this on the PC is both easier and trickier at the same time - switching the PC's date twice will sign you out of Live, possibly making you miss the achievement. Alt-Tabbing to the desktop to switch the date also made the game severely unstable for me, crashing about 3x. What I did was sign into Live then loaded my save at the back door of the courthouse, alt-tabbed to switch to the date, entered the courthouse, talked to CM, exited out the back door again, and after the autosave repeated the process from alt-tab on. When I had 11 dates done I alt-tabbed back to the desktop, used the time synchronize option to make sure the time and date were correct, signed back into live, switched to the last date (Halloween for me) and quickly ran down into the basement and talked to CM. It popped as soon as he hobbled over to the hole in the window.