The explanation for the achievement is self-explanatory. You simply have to log in for 5 consecutive days. You do not have to solve a puzzle, for it to count (thanks to makemeacoffee for confirming). Also double-check that you are connected to the internet on each day, since I'm pretty sure you need to be in order for the achievement to unlock.
The achievement will pop at the starting screen.
For those eager to try: Switching the date on your phone does not work.
Note 2: BUGGY. SEVERAL PEOPLE INCLUDING MYSELF HAVE LOGGED IN FOR 5 CONSECUTIVE DAYS, BUT NO ACHIEVEMENT.
A lot of us unlocked it after 6 days and a few hours. Unlocked mine after 6,5 days.
UPDATE 2: After the update the achievement should unlock immediately, if you met the requirements before. (see comments)