The Pokémon League consists of the building that houses the Elite Four, as well as a building in the western part of the area that houses a nurse who will heal up your pokémon, as well as a pokémart that sells pokéballs of many kinds, along with the standard healing items. There is also a PC here.
This is where you should get ready for the Elite Four. Get a bunch of Hyper Potions, Revives, Full Restores, Full Heals, and any X items (X Defense, X Attack, etc.) that you think might help out.
If you still need to train up your team, you can look for Audinos in Route 10.
When you are ready, go up to the Veteran who is guarding the building where the Elite Four dwell. He will talk to you and ask if you are ready to go in. If not, just say no. Then the next time that you talk to him, he will let you inside. Once you go inside, you won't be allowed back out until you are either defeated, or you defeat all four members of the Elite Four. There are no nurses in there and no way to buy healing items or switch your pokémon, so be sure to buy all of the healing items that you need, and assemble your ideal team of pokémon to take on the Elite Four. When you are ready, go in!