There are any number of ways to play the Pokedex game, depending on where you are and who you're with. You can play it online or off, and even make up your own rules. Since it's not officially sanctioned by Nintendo or The Pokemon Company, you're free to improvise as you see fit. The Pokedex game is a chance for you and your friends to show off your knowledge of Pokemon and can be a great way to pass the time on a boring day.
At its heart, the Pokedex game is about listing Pokemon in numerical order as they appear in the Pokedex. Start with any given Pokemon, then have the next player name the next one in order. If you started with Charmander, Charmeleon would come next, followed by Charizard. Once you get to the end, loop around back to the beginning until you've named them all.
Before starting, you'll want to determine just which version of the game you're using. The Sinnoh Pokedex has a different numbering system from the Unova Pokedex, and the Hoenn Pokedex has yet another different order. The National Pokedex may be easiest, as its numbering has remained consistent across the games, adding new Pokemon as they're introduced. If you choose to use the National Pokedex, decide which versions you're including -- will you only go through the Pokemon from the original Red and Blue versions, or will you try to go all the way up to Black and White?
The Pokedex game is a fun forum game, as long as it's not against the rules. Check with your forum's moderator before starting a round of the Pokedex game, as some moderators might think it's spamming to have people posting only the names of Pokemon.
One fun variation on the game is to have players identify Pokemon by their Pokedex entries. Decide what version you'll work from, then read off the Pokemon's Pokedex entry. Whoever gets it right gets to go next. This variation on the game can be particularly entertaining, as some Pokedex entries are quite strange.