This is easier than you think. Remember, you are talking about getting kids using computers, which is something most kids love. At Community School of Excellence, Aspen Academy, and Lake Middle School we had interested kids before we had a club for them to join. What got them interested? I set up some Linux computers in my room for kids to use. Right away, they liked how the computers looked and functioned differently from what they were used to.
That's a good place to start. Set up a few Linux computers in your room and give kids the opportunity to try them (you will need internet connectivity for this. Talk to your school’s IT department for help). Encourage them to navigate around the GUI (graphical user interface), seeing where your distro of choice puts its programs and files and folders. Let them try out some of the open source programs that come with the distro or that you added yourself. You may even want to include some games (obviously kids like those).
Once kids have had a chance to try Linux, then invite interested kids to join the club you are starting, telling them they will get the opportunity to learn not only how to use these and other programs, but how to install and configure them.
In the past, I have made a point of announcing to all of my classes that the club is starting for the year, letting students know how to sign up. My club has also put recruiting posters around the school to advertise the club, in addition to having an announcement read over the school's P.A. system during morning announcements.
Have interested kids fill out a registration form and turn it in. From there, you build the roster for your club. Kids outside the club seeing their friends having fun will want to join, too. Kids in the club will invite their friends. It takes on a life of its own. In fact, you may find that your members become your best recruiters and advertisers.
Once you have a club roster, you can decide what type of student leadership structure you want the club to have. For my clubs, we decided to have these student offices:
President
Vice President
Treasurer
Secretary
Interested students nominated themselves as candidates for office, and then we had elections. We also have one year term limits on any one office, so more kids get the opportunity to be an officer. There are a lot of different ways to create a student leadership structure, if you think one is necessary. Do what works best in your context.