I know I will get a down vote for this, but I disagree.

I don't want a team to be forced to play a character so that they can get a buff. A secret passive is ok, but something to the scale of Rengar and Khazix makes me feel as if Riot wants to force people to pick Rengar to go against Khazix, and this could even lead to someone not picking a champion they enjoy so that they don't risk giving the other team a massive buff. I think a character based miniquest will wreck the team comps. I would love some Lore based gameplay, but I think its best to limit it to maps or special event games.

Why not, geo? I see where you're coming from, but people at the moment already do that exact thing, play characters they don't want to to counter enemies.

How has Swain vs. Jarven not been mentioned yet?

I want a Warwick VS Soraka!, maybe that if he kills her after a number of lvls, he gets a buff that gives him her heart or something >

Swain vs J4 would be awesome
Idk I think that would make games a lot more fun, and would open up opportunities for custom games too

Probably would be better if this interactions would only take place on a special type of match, a new "lore match mode" or something like that would be awesome actually

I'm all for this, but it'd have to be something 'minor', like the +10 gold if Zilean or Volibear kill each other, or the Kha'zix/Rengar interaction; something fun that gives you a nice buff, but nothing absolutely game breaking or that would force you to go out of the way for rivals to kill one another.

That said, I'm definitely more for cosmetic interactions like Voli's taunts and TF paying out Graves. I love those little rival interactions and vocal cues.