Well, for starters I would go to the web site of the
National Spelling Bee
as sponsored by Scipps-Howard. From there you can find a
list of last year's contestants that includes sponsoring newspapers.
The Pittsburgh Post-Gazette is one, but there may be another even closer to you.
They should be able to provide you with information about regional qualifying events that could lead your champion all the way to national finals on ESPN. They can also help you obtain copies of the official word lists and study guides, and provide information about how to organize a qualifying school event.
That said, every school I've been involved with has started with classroom level competitions, with the number from each classroom partially determined by how big you want the school championship to be. I've seen it with 1, and I've seen it with 3.
Usually each classroom teacher has passed out the official school competition word list for studying, and then lines up the kids and asks them to spell a word. The teacher should use a standard pattern to select the next word (each in order, every seventh word, whatever) to make sure there can be no claims of favoritism when the one student gets "dog" and another gets "nom-de-guerre". Get it right, you're still in; get it wrong, you're out.
If going for a single champion, the standard format when down to two at least used to be "spell the word the other finalist missed PLUS an additional word". (In successive years at our school, the additional word was "veterinarian" for a girl who's father is one, and then "argumentative" for a boy who's father is a lawyer).
If you're looking for the top three, you need to make sure that one person doesn't get in just because they're at the end of the cycle. If you start a round with four kids left, and the first misses their word, the other three should not automatically go on. All kids need to get a word. If there are too few standing after the round, those still standing go on; those eliminated in that round are brought back for another round and repeat until you get to the correct number of finalists.
The school championship needs to be a contest to a final winner. We always start with one round of "cat" and "dog" words with no elimination just to make sure everybody is comfortable with the format. Kids with previous experience will be given words like "pair" or "pear" so they will ask questions like "definition please" or "used in a sentence please". You'll want one person reading the words, and two judges checking the answers. If your school has a speech therapist, they make a great choice to read the words; I've also seen it be the librarian and the principal.
Good luck, and let us know how it turns out.