I'm sorry, I'm not entirely sure what you're asking for. If you want fanfiction based around Jax, I'm afraid I can't help you here, you might want to ask on Concepts and Creations. I can't help if you want official info either, since I don't work for Riot (and even if I did, I wouldn't drop the Urf icon). But if you want a too in-depth, long-winded, dull infopost about analyzing his character...
Okay so let's start with the old Lore. Jax's Lore is in a really weird state right now because he's one of the champions most drastically effected by the retcon to the Institute of War. Jax's old Lore was so filled with connections to the High Summoners and the League that he's sorta got a gaping hole where that used to be. It sounds like you're already familiar with all that, so I won't bore you with the details you already know, but I will hit you with some classic old Lore forums silliness, we used to talk about who Jax was all the time. The most popular candidate, easily, was that Jax was Reginald Ashram, the first High Summoner, who left for...reasons, and came back. Note that the doors to the League swing wide open without needing to test him at all.
That's all in the past now, though, so focusing on his new story, we can get a lot more canon information on Jax, especially regarding what he is. Jax references being "homesick for a land that no longer existed", regarding Icathia, which a lot of Lore fans will know has very strong ties to The Void. He's also mentioning trying to find allies capable of fighting "the monsters from beyond", who we can pretty easily presume are the Voidlings.
Jax being from Icathia is HUGE, because that means a few things. He's probably extremely old, for one, if he's homesick for a forgotten city. Whatever Jax is, he's either got an incredibly long lifespan or he's no longer aging. That leads pretty well into the second point, what even is Jax? If you take a close look at his splash, you can see three fingers and a mask that seems to contain five eyes (though this could be like a Master Yi thing, he might just have a weird mask). He's also got purple skin. He's clearly not human, but I think the purple skin is the best clue. The Void is often associated with purple, and given Icathia's affiliation with it, as well as something similar happening to Kassadin, who also was exposed to the Void, I think we can guess that the Void somehow altered Jax a bit. That's not totally canon yet, though, just a (somewhat) educated guess.
Personality-wise, Jax loves fighting and combat, but not in the bloody sense. He doesn't seem interested in killing anyone, and actually goes out of his way not to grievously harm the Demacian soldiers attacking him. ("The metal buckled and the man went down on one knee with a grunt of pain. Jax gave him a moment to still the ringing in his head.") The reason he uses the lamp, along with other unconventional weapons, is to limit his own strength, which is carried over from the old Lore as well.
He's also seeking worthy opponents, possibly just for fun, but as his new Lore seems to hint, also maybe to find allies to fight back the void. Fiora seems to be a candidate there, judging by his new short story, and given their friendship in the old Lore and him still being listed as a "related champion" to Jax, I would hedge bets towards Gragas perhaps joining his crusade. Remember that not even Aurelion Sol could take the Void on at his full strength, so this says a lot about how cocky Jax is, and also just how good he is.
This isn't exactly part of Jax's lore, but I always feel like bringing it up because it was sweet. Jax was the favorite champion of a kid named "Joe" who played League and had cancer, and went to Riot with the Make-A-Wish program. For a limited period of time, Jax and Jaximus (Joe's favorite skin) were made half off in the store and had their proceeds donated to charity. Sadly, Joe did pass away from his cancer, but Jaximus was given a new line: "Here's looking at you, kid." in memorial, by Jax's VA, Adam Harrington, who is seriously rad and contributed a TON of voices to early LoL.
(also Jax might be a lil' pudgy, or maybe it's just more weapons hidden under there.)
this thread & say GIVE HIM A DECENT SKIN PLEASE. Its been well over a Year & most of his current obtainable skins are shit.