Edit: This achievement is pretty easy to get and you should get it without too much effort. This guide is mainly here to explain how invitations to all clubs work.
To get invited to a club, you need to draw their attention.
There are four clubs; the Hunting Society, the Frontiersman, the Boston Brawlers and the Thief's club.
As of yet, the requirements for all are a bit unclear as to how much is needed, as you can fill the requirements in different ways. For the Hunting Society you need to hunt, for the Frontiersman you need to explore, for the Brawlers you should fight with your fists instead of weapons, and for the Thief's Club you need to pickpocket. It's not enough to just make headway into the club's challenges, for instance I had already completed many level one challenges for the Brawlers, but the invitation will not be given untill you've fought with your fists for a bit.
You can start on these requirements once you start playing with Connor.
The easiest ones to start with (to get the achievement) are the Hunting Society and the Thief's Club. If you follow all optional goals in the third mission of sequence four, you should be almost done for the Hunting Society. It unlocked for me at the start of sequence five, when I encountered my first cougar (encountered, not killed - that meant I encountered all game from Kanièn:keh; hare, fox, deer and cougar).
For the Thief's Club, you have to pickpocket at least £ 200. Note that there are other ways to aquire an invitation for the clubs, for instance I got the invitation for the thief's club after pickpocketing only £ 105, but I had also caught couriers, robbed corpses, and performed aerial kills.
Out of the comment
moogless left and my own game combined, I've come to the following conclusion: Once you've explored enough, Frontiersmen will appear on your map (the symbol looks like the one for the frontier itself). Talking with them starts side-missions; once you've started you first one of these, you should get the invitation to join.
I got my invitation to the Brawlers after fistfighting my way through a convoi (for the Prince Of Thieves achievement). I hadn't fought much with my fists before. After meeting the requirements for the Brawlers, a man will walk up to you and invite you for a tournament to prove your strength (it doesn't start immediately, you just unlock the possibility of starting it).