1. I'd suggest the same methods used for most activities... paper ads on campus walls, suggestions made in class by you or other teachers, and talking to bright students (top half of the students in academics) who may be interested to provoke inquiry in them about possibly playing. Once you have chess players, encourage them to spread word of mouth advertising.
2. I think there is a lot of value to having an outlet for unrated, casual chess. On the other hand, I think you will want chess clocks to encourage quick games. Also, I'd suggest a chess tournament every 3rd meeting or so. Divide it into two sections and put 4 or so players identified as the best in the first section and the rest in the second. Every time a student wins the second section, give them a certificate and move them to the first section. You may, of course, develop an inhouse rating system for these games, but...
3. Encourage students to play online, either on school computers (as permitted) or at home. Have them register for matches against each other online. Encourage the use of a Java-based computer opponent at a slightly higher skill level than the student for increasing chess ability. For those who are already taking a real shine to the game, give them information on how to play in an honest-to-goodness rated USCF tournament. Once hooked on the idea of getting a good rating there and placing in events, they'll be players for life.
About puzzles: It would still be a great idea. I'd suggest giving a bit of a clue as to what you want from the student and how fast you expect it, in order to encourage them to hunt for it (even if that is slightly artificial). So "mate in 2" or "material gain in 3" would be good captions for puzzles.
I'm a high school teacher and have ran a general board game club for about 3 years (2 of those years at a middle school and this last year at a high school). So far the club has primarily played various non-traditional board games.
Last year I focused on the game of Go and organized a competitive team to play in online tournaments. In the first year we were able to do decently in the one tournament we entered. Unfortunately when I made the move to the high school the Go team kind of fell apart.
Now, I am intending to do something similar and organize a chess program at my school. My intention is to have it meet at the same time my strategy game club meets due to me having limited time during the week each week. I intend to mark off part of the room for chess only in an effort to keep the chaos of the other games a little separated.
My questions for other sponsors are on a few topics:
1. How do you advertise within your school for your club/team?
2. Do you do rankings within your club (seeding like a sports team does)?
3. How do you keep up interest and encourage students to improve on their own?
For my previous Go team I would find puzzles and post them on a bulletin board each week. Students could choose what they thought was the solution (normally a single move) and drop their answer into a box. The next week I dumped out all answers and everyone who got it correct got a little piece of candy. Would something like this work for chess, or are all chess puzzles/problems multi-step?