Eliminate the Act IV final boss - Diablo - while accomplishing certain tasks during the fight. They can be done over as many playthroughs as you want.
Uncaged - Kill Diablo without getting trapped in a spike trap and without dying on Hell difficulty or higher
While fighting Diablo, she will sometimes have glowing hands where it looks like she is casting a spell, which is your queue to roooooolllll the hell out of the way. This can be very tricky to dodge, and may take some practice, but having allies to help distract or stun her can beneficial.
Self Sufficient - Kill Diablo without using any health globes or healing wells on Nightmare difficulty or higher
There are a couple of healing wells located in the main boss room, and she will also drop health globes from time to time during the fight - avoid both. You can still use potions and Monks heal spell if needed. This is another example where a coop team to speed things up can make life easier, so long as they also know to not pick up any orbs (which will also heal you).
There Can Be Only One - Fight and kill 5 clones of yourself in Diablo's Realm of Terror on Hell difficulty or higher
During the 2nd rotating phase, you will be in a shadow-world where you must fight clones of yourself, and also deal with Diablo when she appears. This will take a couple of attempts, as you will only see 3 or 4 of your clones.