You get a star for completing each of these objectives:
-Defeat the Elite Four
-Collect 5 Shiny Leaves on one Pokémon and speak to Ethan/Lyra in New Bark Town
-Defeat the records on all 10 Pokéathlon events
-Get a 100-Streak in the Battle Tower
-Complete the National Pokédex. You need to capture all Pokémon other than Mew, Celebi, Jirachi, Deoxys, Phione, Manaphy, Darkrai, Shaymin & Arceus, giving you a total of 484 in your Pokédex.
This is straight from Serebii.net. Source down below.