There are two ways to become part of a Club: join an existing one, or create your own.  Both of these can be done in the Club menu in the lower-right corner of the display.

Join a Club
To join a Club, scroll through the Club list and select the one you wish to join.  Alternatively, use the search function to find a specific Club you're interested in joining.  Open Clubs can be Joined right away, whereas closed Clubs can be joined by request; you will become a member when the Club leader accepts the request.

Create a Club
To create a new Club, select the 'Create' icon at the bottom of the display and enter your new Club's details in the window that appears; this determines how your Club will look to players searching for it.

It costs 500 Coins to create a Club.  The player that creates the Club automatically becomes the Leader.