By far, the most important feature of a video game club is the teacher who runs it.
Just like classroom teaching, above all other factors, the most important feature to a successful club is the person running it. They set the tone for the club. If your club isn't as successful as you're hoping, first look at what you're doing before looking to see what parts of the club you can change.
I will be recommending different leadership approaches for running your club. However, in the end, this is your club and you should run it however you feel most comfortable. These are merely suggestions of best practices that I've seen other people use successfully in their clubs.
Own the club
This is your club. You must put forth your fullest effort into it, or else it will sputter. Do not take running an after school video game club lightly. This is not the kind of thing that you can just set up for the kids and let them do their thing while you grade at your desk after school. Get in there, play games with the kids, talk to them about the games, let them teach you about them, go home and research into the games that you don't know about so you can come in next week and chat them up on a more advanced level. These kinds of clubs take work to be successful. Pour your heart into it.
Your biggest priority needs to be the students
Spend the majority of your efforts getting to know the students and build a relationship with them. You can have a perfect set up of brand new consoles and every game ever created all connected to 4k projectors in a theater, but your club still won't be that successful if you don't put your effort into the kids who come. Get to know them, give them a space to let their façade of the normal school day melt away, relax the rules a bit from the normal school day, encourage them to take risks, mentor them when needed, introduce the kid who sits by themselves each time to the other kids. Running a video game club is not a passive endeavor, you need to engage with them.
Constantly look to improve yourself
Never be satisfied with your club, even if it's going quite well. Always look for places to improve upon. This will not only make you a better leader, but it will also have the side effect of keeping your club feeling fresh. Your club will feel your changes and change as you do. It'll keep things exciting. This kind of personal work can range from how you, as an extrovert, can improve how you connect with introverted students in the club; or it can be related to club business, like wanting to expand your club to include the school's sports game fans. Feel free to rejoice in how well your club is going, but never feel satisfied with keeping everything the same.
Don't feel afraid to give the students tasks within the club
Practice letting go of the controls. Let the kids run the tournaments, or plan what will happen at the next club meeting. The best thing is that the kids get invested into the club, and there's no better way than to let them control the parts that mean the most to them. It's perfectly reasonable to guide them in a direction you think is healthy, while they decide what to do. Also, the more the kids are invested in it, the more they'll keep coming back, and the more they'll tell their other friends to come.