This is gonna be extremely opinionated but here me out.
Disclaimer I don't main the champ so it's not "Well you're being bias" because if that were true I would say Rakan because I like Rakan. However if I'm being honest it's not Rakan.
Let me start by explaining my reasoning on how this champ could earn the title of champ with the best lore based on what we know or whats been hinted at.
First I have to explain what I feel makes lore stand out above other lore.
1-Personality, pretty self explanatory, though Rammus has a lot of hypothetical intrigue 'okay' doesn't constitute personality. Be it good or bad unless it's a part of your character that you explicitly don't have personality it's a key to good lore for me.
2-Motivation and/or a cause. Also self-explanatory, a character with no reason to be doing what their doing isn't interesting to me and so champs like Ivern who is doing who knows what for who knows why aren't gonna make it. Lucian is a good example of a character with good motivation, but outside of a few stories he is kind of bound to revolve around the same quest with no real growth. Then there's Shen on the opposite end with a cause to fight for, but with all that's happened he doesn't seem to have a driving motivation, something he works for, not something he works towards, if this were added he might've been my pick.
3-Strengths and weaknesses, character flaws, and inner struggles. For example, Braum has a lot of strengths but he has no real struggles or weaknesses really. Yasuo would be a good pick because of his personal struggles with guilt and his inability to accept forgiveness, but he doesn't take the spot because though he has many things here he misses a more important aspect.
4-Connections. A really good example of a champion with a lot of connections is Ryze, and he is probably the champion I would put directly after my choice purely because in the present though Ryze has a lot of connections as seen in the cinematic, there isn't much depth unless you look at his lore with Brand which is his strongest part in this. Darius is a really good example for this because of his position in Noxus, his relationship with Draven being fleshed out in his comic and lore, and he also has a lore connection with a child and its mother in the comic being that he was the father (allegedly but I don't know) however Darius' motivations are largely puppeted by Swain so he can't stand out and be the number one.
So who is it?
Garen Crowngaurd. Now before you throw your keyboards and rage at the screen for the audacity that I have demonstrated to insist that Mr.Spin2Win could possibly ever have the best written character in the whole of league of legends if a little more confirmations and development comes for him, hear me out some more.
**Some minor spoilers for LUX may happen here, be warned**
1-He has a fleshed out personality, both in game, in the stories, and in the comic. He's honorable, confident (even a bit arrogant), cares about his families name and is protective of his family as seen in the Lux comic where he rushes in to find Lux after the explosion.
2-He's motivated to bring justice, spread the glory of Demacia, which is the cause that he fights 'for the glory of', he's also motivated to keep his family safe, and bring down evil.
3-While he is strong physically, Garen is oblivious to the nature of petricite, struggles with his sister being a mage while still having to maintain the laws of Demacia and preserve his family name, and as stated above he is arrogant. Not to mention Sylas literally incapacitating him after a fight.
4-Connections, not only has the Lux comic and their shared story shed light (hehe) on their relationship but it shows that Garen is caring for his sister and works hard to protect her, often unknown to her. The comic also expanded upon his closeness with J4, his interactions aunt and the guards, and shows a rival in the form of Sylas combating ideals. Not only that but he also has a rival in Darius, and many sources (pantheon, tahm kench, previous lore, his own voice lines) all point to him and Katarina either having or having in the future romantic interactions. Which should that be explored more deeply in future lore will develop him even more, first being changed by experiencing a mage rebellion caused by your sister, and then falling in love with a sworn enemy of your nation, of which he is incredibly renowned for being the leader of the Dauntless Vanguard.
So there you have it, agree, disagree, leave a comment below telling me what you think about my opinion.