Joining the league is simple. Just follow these steps.
First, review the rules for selection of Pokémon and Squads.
Secondly, once you understand what you are allowed to have in your squad (please, don't hesitate to ask a League Official if you are unsure), create a squad of up to 15 Pokémon and post it in sigless squad submissions. Pokémon posted in sigless submissions are considered automatically approved, so long as they're allowed at your trainer level - post them here and you can start battling straight away.
At any time after adding a Pokémon, you can create or change their signature move by posting it in squad submissions. Your signature moves will be reviewed as soon as a League Official is available to do so and will either be approved or suggested modifications given. For moves where there were suggested modifications, you can either accept these, in which case the move is approved, or create a new move for approval. If you make changes besides the ones the League Official has suggested, then you will have to let a League Official know that changes have been made. There's a short guide on the format of how to post a squad in the main threads forum that you should read before posting. There's a formatter created by Rabscuttle to automate this process, which can be found under the main menu of the PASBL site.
Please note also that it is a cardinal rule that you are NOT allowed to edit your posts after a League Official has touched them if you are not needed to (e.g. In the case of squads, if all your moves were approved, you editing the squad could possibly mean (And has before) that you edited your moves to give them more power and appear as if they were approved.). Consider this your first and final warning.