Joining the League
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 and post it in sigless squad submissions. Pokemon 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
Finally, once you have created your squad, post it up on the thread entitled "Squad Submissions". Your signature moves will be reviewed sometime in the next week, (Usually much sooner, however there ARE sometimes busy periods where no league officials may be available.) 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.
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.). People violating this rule can be expelled from the League on the spot if it is deemed necessary. Consider this you first and final warning.
UPN Squad Submissions
SPP Squad Submissions
SPP Sigless Squad Submissions