"Play as Batman and send all Gotham villains Through the Arkham Transitions"

The Batman Villains are:

Joker
Harley Quinn
Bane
Deadshot
Scarecrow
Poison Ivy
Catwoman

Go to Single player > Single Fight, Chose the Arkham Asylum Map, Pick Batman, Pick one of those villains, chose any difficulty, make the Transition.

"Harley must dethrone an opponent in King Of The Hill with the final hit coming from Double-Trouble"

To use "Double-Trouble" you need to hold the button of Harley's Character Power.

You can spam Harley on KOTH till you get it.

You can also ask someone to help you, if the one that is helping you challenges you for KOTH he will be the King.

"Win a match with each villain"

You can go to Single Player, Single Fight, Pick the villains and let the A.I. play for you.

Or you will get it eventually.

"Raiden must summon Kidd Thunder during a match"

It's a victory pose (it happens after the enemy's first health bar).

Win with Raiden till you get it. You can let the A.I. fighting for you, eventually you will get it.

"Be in the Batcave when all the Environmental Interactions are used"

Go to Single Player, Single Fight, Pick your Character, chose a loadout that is not too strong, pick an enemy, chose the Batcave Map, use all the interactions possible.

If you are having trouble, try once with Batman and once with Superman, since they interact with the map in different ways.