Gen I Indigo League:
Just in general we know a bit about them as the Fame Checker outright tells you trivia on them. Kind of wish all games had the Fame Checker to tell you bits of information like that, but alas they don't and some Elite Four really suffer for it.
Lorelei: Was given a minor role in FRLG. Lorelei is from the Sevii Islands, Four Island specifically. She's a logical, collected, and cool person; fitting her Ice-type preference. Yet she also likes to collect Pokemon Dolls in her home on Four Island, showing some more layers of her personality. She's protective of her home region including the Icefall Caves where as a child she met her Lapras.
Bruno: Notable for being an Elite Four of two different gens (Lance moving on to being the Champion in Gen II), the main thing anyone remembers about Bruno is his love of Rage Candybars. Yeah, Bruno really doesn't have much depth put on his character. He's a tough looking martial arts guy and the only other info provided on him was how he likes incorporating spas in his training and he once trained with Brawley.
Agatha: To me she has the most interesting backstory though they never delve into it. When she and Professor Oak were young trainers they were fierce rivals (and maybe there was some hinted romance between the two, however obviously it didn't go anywhere as she's not stated to be Blue's grandmother). She had a hot temper and short fuse. She's also apparently the oldest Elite Four member, though that was before Bertha was made which even the game has noted they she has similarities with.
Lance: Becoming Champion increased his presence shown by his role in Gen II. Growing up in Blackthorn, Lance is cool and passionate and he's not afraid to make a statement like hi preference for capes and stating that Dragon-types are indestructible. FRLG seemed more interested in his family relations with it noting Clair is his cousin and the Elder might be their grandfather. In Gen II he's presented as a sort of action G-man, working on behalf of the Pokemon League to stop Team Rocket's experiments. But he still show he cares like him defeating Silver and scolding him on how his behavior which led Silver onto a better path, and shows no anger toward him when he has the the double battle with Clair against him and the player (he even apologizes for maybe being too harsh).
Gen II Indigo League:
Will: He... exists? Only notable thing about Will is his clothes which just makes you want to know about him. He goes on about being a psychic an having limitless possibilities but that's all and is kind of generic talk for a trainer. Is he a performer of some sort? How does he stand out among other psychics we've seen? At the very least give us a personality quirk!
Koga: Koga has a bit of info from his thanks to being a Gym Leader in Gen I (not that any other games count that as BW2 showed). Koga is a ninja master which translate to his preferred battling style is inflecting status ailments (poison, paralysis, and sleep). However his expertise in poison has also taught him about medicine which he uses to heal his Pokemon. He also actively patrols Fuchsia City and the surrounding areas including the Safari Zone. Everything else about Koga is about him being the father of Janine.
Karen: Everyone knows Karen because of her philosophy but she's just as unknown as Will. She not only has a plain appearance (yeah she's hot, but so are most other female trainer classes) but her only character trait is treating everyone equally and likes the Dark-type persona. So how does she relate herself to the Dark-types beyond that? What does she think about trainers who disagree with her philosophies which many do? Does she like long walks on the beach? What else do you got on her?
Hoenn Elite Four:
Sidney: It's said he likes showing off the beautiful side of Dark-types but nothing is further addressed about that. He's a punk gone to Elite Four and has a "might is right" way of doing battles but that's it. We see him in the Battle Maison in ORAS so you might think you'll learn more about him but he just says he was there looking for Steven. Come on! At least say he likes battling there or maybe he doesn't. Like what are his views on the Battle Facilities now that he has appeared in one? Oh, he's just here for Steven but he missed him so now he's leaving. Fine...
Phoebe: She gets a bit more to her character. Her grandparents are the elderly couple on My. Pyre protecting the Orbs. From a young age she was able to talk with ghosts (not just Ghost-types, ghost ghosts as well) yet has a pretty happy-to-go attitude. Her additional scene in ORAS shows all this with her happily talking to her grandmother among tombstones... except you don't see her grandmother which implies she had died during the post game.
Glacia: She's not from Hoenn but a colder region. She came to Hoenn in order to train her Ice-types in warmer weather. She has "flaming passion in icy cold" and often make similar heat metaphors. Nothing else though, the only other notable thing about her is that she's the only member of the Elite Four in ORAS not to get an outside of League cameo (though its mentioned she likes to eat at the Mauville Food Court).
Drake: He's a sailor who train Dragon-types after being saved by one. He has respect for Pokemon being free wild beasts who may help or hinder, saying you need a virtuous heart to battle alongside Pokemon. He also doesn't seem to think too highly of the Emerald Battle Frontier saying if the player wants a real battle there's no substitution for the League. He's also friends with Mr. Briney and Peeko.
Sinnoh Elite Four:
Aaron: He REALLY like Bug-types. Moving on... WAIT! There's maybe a bit more. BW2's Variety shows talks about a Sinnoh Bug Catcher who went on to become a Gym Leader and then a member of the Elite Four. So was Aaron a Gym Leader at some point? Well that's all that's mentioned on that point so now moving on...
Bertha: It's not really Bertha herself which is focused on but her likeness to Agatha. They're both old woman who look identicla, their names are similar, and their attitudes seem to purposely be opposite of one another (Agatha is rash and quick to anger while Bertha is gentle and sweet).
Flint: One to get the most info about here and that's still barely anything, he's friends with Volkner and when Volkner started acting distant and maybe thinking to quit being Gym Leader it was Flint who asked the player to battle him. He's also the only Elite Four to show up in the Villa in Platinum where you find out more about his quirky attitude like saying since you bought the furniture with the prize money you won from him he gets to lounge there, says Fire-types are better than Electric-types (further building on his friendly rivalry with Volkner), wants to add some more decorations to the league, and its mentioned his fascination with Fire-types is because of his name. I also could have sworn he said he had thought about hiding his Pokeballs in his afro to surprise trainers.
Lucian: He likes to read and that's mainly it. In Platinum you can find him in Canalave Library occasionally and he'll talk about the book that he has been reading but not much else on him specifically. From the comments he makes him seems a bit of a passionate philosopher or at the very least likes to read about that subject.
Unova Elite Four:
Shauntel: A novelist, she has written many books and is constantly writing another, often making her absentminded to what's going on around her. By the way she writes you can tell she's analytical about details. She also cares a lot about her Pokemon, apologizing to them first when she loses before addressing the player on their win. She's also a bit quirky, basing her clothes (or at least her collar) on an animal (a black cat if you couldn't tell).
Marshal: Marshal is the apprentice of Alder so he gets a bit more roles in the story as Alder asks him to help do tasks such as guarding the entrance to Twist Mountain in BW2. Marshal is very dedicated and values strength and learning from your battles, win or lose.
Grimsley: Once from a distinguished and rich family, they fell to ruin for some reason. Because of this he became a gambler (and also doesn't wear socks) and believes that a win is a win no matter what.
Caitlin: A former Frontier Brain, her story starts in Platinum. Though the Frontier Brain of the Battle Castle in Platinum, she never battled but instead her butler Darach did with her overseeing them. The reason for this wasn't revealed until BW when you find her as a powerful Psychic Elite Four member. As a child she would have tantrums whenever she lost battles and her powers would go wild (you can see some bit of this as she tells Darach to give the player Castle Points "At once!" upon winning). So to prevent that Darach battled in her place until she have matured. By then she moved from a Sinnoh Frontier Brain to a Unova Elite Four member, coming to Unova to hone her power.
Kalos Elite Four:
(Not going to cover them that much since we're still in Gen VI)
Malva: A member of Team Flare, she's very passionate about their goals seeing it as necessary to do. But we don't really find out much about her, just her loyalty to Team Flare. She's easy to anger and isn't afraid to threaten you though nothing comes from it so either she's bluffing or knows she'll achieve nothing in doing so. She's also concern with the intriguing of Team Flare, one reason for her agreeing to help the player and Looker arrest Xerosic is because she believes he's using Team Flare resources to experiment on a child (she totally okay killing them blindly, though). She's also the Holo Caster reporter, providing her more ways to help Team Flare by creating cover up stories for their activities though otherwise tells the news as is.
Siebold: A passionate chef, he sees the art in everything if you put a bit of skill into it be it cooking or battling. Anyone who disagree with him is met with him scolding them for being a bit close minded.
Wikstrom: Has a very proud and chivalrous attitude, even speaking in ye olde speake. He looks forward to having a honorable match against the player and making a dramatic scene whether he wins or lose.
Drasna: Her grandparents are from Sinnoh having grown up around the myths of Dialga and Palkia leading to her fascination with Dragon-types. Otherwise she's just a kind elderly woman.