What is there to do after becoming Champion?

Not currently aware of a full list, compiling one below. Possible spoilers and bad jokes.

Collect all the Cells

Zygarde collection quest. You can finish collecting the Zygarde Cores/Cells, some of which are only accessible after becoming champion.

Help the International Police

Lookers Quest. Looker needs help dealing with the Timey-Wimey Ultra Beasts, and Hau's not going to do it.

Gather Island Guardians

Catch the Tapus. Its not like they would be saving people if they weren't trapped in your PC box.

Explore Poni Island

Much of Poni Island is inaccessible until you are champion. This includes such diverse locations as Poni Grove, Poni Plains,Poni Meadow,Poni Coast,Poni Gauntlet, and ~~Poni~~ Battle Tree.

Battle Tree

Battle on a tree. Face off against trainers including the legendary Red and Butts, I mean Blue. Get a 100 win streak to win a ... berry? Or get the full 200 win streak to get a different berry.

Visit GameFreak

Battle Morimoto. Once per day you can take on his arbitrary selection of Generation 1 Pokemon.

Complete the Pokedex

Arguably only completable in postgame due to legendaries. If you're not completionist, you won't be able to get the Shiny Charm for that sweet 0.073% shiny encounter rate.

Battle Royal

Compete in Battle Royal. Could be done before the beating the League but you didn't do that. Rage at the AI for stealing your win right before the end of the match. Win totally functional ribbons for completing Master rank.

Eevee Quest

Earning Eevee's excessive Extreme Evoboost. You get Eevium-Z from a trainer in the Thrifty Megamart on Royal Avenue after beating the eeveelution trainers.

Mega Quest

Mighty Morphing Mega Pokemon. You get the Keystone for mega evolution, as well as Alakazite after beating Dexio on the Ancient Poni Path. Now try to find all the Mega Stones! Try at the Battle Tree, and yes you have to buy them.

Take on New Challengers

When beating the Elite Four in the Post-Game, you will face one of 10 random Challengers as your fifth match. They include Hau, Sophocles, Ryuki, Gladion, Molayne, Plumeria, Hapu, Faba, Tristan, and Kukui.