Why are they mad? A look into the motivations of angry, shouting champions.
With the writing of Aatrox's lore (note: not re-writing; the Darkin weren't a solid idea before), there have been a number of complaints about him being changed from a quiet, mysterious character to an angry rampaging demon. And while I won't go into detail about how the mysterious character didn't go with his visuals, playstyle, animations, or abilities, I want to point out his and other champions' reasons for being angry. I hope these short explanations shed at least a little light on the mindset of these characters.
In order of my opinion on the magnitude of their reason:
First things first, as for the "angry for the sake of being angry" nonsense I've seen claimed, Rek'sai is the only example. She's angry because she's motivated entirely by predatory instinct. She, unlike Rammus, is an animal in body and mind.
A sociopath. He isn't just telling everyone to get off of his property as a joke; he truly believes all of Runeterra is his. He defends "his" land from anyone in sight with all the ferocity of an old coot with a shotgun who thinks you're stealing his peaches.
Revenge for his clan. Simple enough.
Speaks for the trees, and the trees are not happy to be destroyed by magic.
Driven mad by his incarceration.
Unwillingly brought back from death into a body that has no feeling, with little memory of the man he once was.
His body was torn apart and reshaped into that of the monster we see in game, anesthetic intentionally withheld to drive his sanity further away.
And then there's Aatrox:
Following his descent from a godly hero of justice, he was sentenced to a fate far worse than any hell I can imagine. Instead of the mercy of a quick death, he has endured millennia with no body and no senses of any kind.
As you can tell, I ranked it mainly based on the pain which caused their anger, from the complete lack of pain present for Rek'sai, to the emotional pain of Tryndamere, to the intense physical and mental pain of Warwick. Aatrox stands at the top because he has the most reason to be angry: Loss of a clan? How about an empire. No feeling? At least Sion can move and see. Physical pain? The body of every host rejects him to the point that the blood, bones, and muscles tear away from each other.
In conclusion, champions are not made angry just for the sake of having an angry character. They're made angry because it fits the story.

He's hungry all the time and also hates humans. Fair enough
She is somewhat shouty and angry. She thinks everyone hates and wants to kill her because she's too powerful.