Eliminate the Act II final boss - Belial - while accomplishing certain tasks during the fight. They can be done over as many playthroughs as you want.
Good Eye - Kill Belial without dying and without getting hit by his projectile or meteor attacks
During the phase against giant Belial (with the green background) he will try to smash you with his arm, shoot lightning bolts at you, and spew nasty stuff at you. Most of the fight will be spent avoid green circles that appear on the ground that signify where he is about to hit with one of his attacks, and you will also want to avoid his spew attack by moving to his far left or right side to have the highest chance of survival (this can be death at higher levels).
Fist Missed - Kill Belial without getting hit by his fist attack
During the Giant Belial phase, single green pulsating circles will appear on the ground to signify where the boss is about to smash his fist into the ground. Simply avoid these circles to avoid getting hit, and to complete the challenge.
Why'd it Have to Be Snakes? - Kill 10 Veiled Sentinels in the first 30 seconds of starting the fight with Belial on Nightmare difficulty or higher
The Veiled Sentinels are the Serpent enemies you fight at different parts of the fight. To complete this challenge, go all-out as soon as the fight starts to kill at least 10 of the first group of Sentinels that spawn.
Bad Breath - Kill Belial without getting hit by his breath attack on Inferno difficulty
During the giant Belial fight, he will occasionally rear back to spew a green smog at you that hits a cone of about 165 degrees, meaning you will want to be at his absolute left or right when he does this to have the best chance of avoiding it. There is still the offchance that he will lean a bit more toward your direction, meaning you get hit anyway, but this is your best chance.