The quest for this achievement is unlocked from the start.
At the moment it is only possible to join an alliance if either the alliance invites you, or when you select one of the players in an alliance, and you send an application via his profile.
Easiest way though is by going to alliance in the main menu, then create a new one. Pick a name that's between 6 and 16 characters long (and not taken already). You can delete the alliance after unlocking the achievement, by going to alliance -> members -> select your own name -> leave alliance.
To unlock the achievement, go to achievements (after you've met the requirement) and take your reward.