Get administration on board.
Most schools have a policy that new student organizations need the permission of an administrator (perhaps a specific one). You might find that your administration is a little nervous about permitting an improv troupe - the unscripted nature of teens on stage can be a little...unsettling. Let the administration know that there's a lot more structure to it than it might seem.
Establish clear standards and expectations.
It can be very easy for improvisers to simply go where their performance impulse takes them - but at the high school level, there are certain lines about sex, drugs, behavior and so forth that just might not be appropriate. Depending on where your school is located, those lines might be more or less strict. Get all your interested kids on board with the notion that these aren't flexible. Determine what type of show best suits your local audience, and then emphasize that in rehearsals.
Decide what kind of improv you want to perform.
The easiest entrance into performing improv is short form: improv games and scenes very much like Whose Line is it Anyway? It tends to be easiest for teens to get their heads into, and they contain fairly easy gimmicks to make them funny. Some improv purists argue that doing short form improv makes you actually a bad improviser, but that's debatable.
Another option is to establish a troupe dedicated to long form: scenes which run much longer - even up to a completely improvised two-act play - in which characterization and storytelling are much more important and integral. Starting a long form troupe from scratch - with students who don't have any improv background - can be more difficult. If you have a group of solid drama students who have inclinations towards improvisation, you might be able to pull it off; long form tends to rely much more on acting ability, group memory, and storytelling ability.
Decide who the coach will be.
There's no reason why a drama teacher with improv experience couldn't be the coach of a school's improv troupe - but some teachers would rather bring in outside help. If you have any local improv troupes, either professionally or at nearby colleges, you might consider bringing them in to coach the troupe. Make sure that the coach you choose can commit to it regularly - if you have a professional who can help but can't commit long-term, then perhaps a better choice would be to have a drama teacher serve as the coach/sponsor, and bring the professional in early to do a number of workshops.
Set a clear rehearsal schedule and structure.
What day of the week will fit with the extracurricular schedule? Some schools stagger how their clubs meet, some leave it wide open for clubs to choose - but in that case, you'll need to consider conflicts with other organizations. Once you know the day - and how long your troupe will meet on that day - figure out what rehearsals will look like. How long will warm-ups take? Will the coach talk a bit at the troupe, or just jump right into scenes or games?
Unlike normal drama rehearsals - which is structured specifically towards scripted material performing on a certain date - improv rehearsals tend to be more skill-based: pick a particular type of improv skill, and work on that for one rehearsal.
One final important note about rehearsals: you absolutely must allow time/opportunities for coaching - breaking down what worked/didn't work in a scene, critiquing player's performances, etc. Without it, rehearsals just won't be effective.
Find your rehearsal space.
Ideally, if schedules accommodate it, you can rehearse in your school's performing arts facility. The nice thing about improv, however, is that with a lack of sets or props, rehearsals can take place almost anywhere. The best option is to find a space that is easy to reconfigure to give you plenty of space to play, a clear delineation of stage and seating, and somewhere you might be able to get loud. Classrooms are just fine, as long as there's room to get desks out of the way. If possible, make it a consistent rehearsal space, so that troupe members have a familiar, comfortable space in which to open up.
You can't have a troupe without performers. Advertise auditions in the school theater, in drama classes, and around school. Don't just stick to the drama kids - you might find that students who have a talent for improv might not do well or fit in with regular stage performing, so they may be "hiding" elsewhere.
It may take a while before your troupe (or you) feel ready to perform on stage in front of an audience. As your troupe gets to that point, you may want to take a look at our tips for improv shows.