how to define "toxicity" is very simple. Champions like: Yi, Darius, Teemo, Heiemerdinger, Jarven etc have probably all been considered toxic at one point. Toxicity is not "overpowered omg" though that feeling can occur. I would define Toxicity under the following:
Champions that are built upon punishing mistakes (Darius). New players make mistakes, lots of them. With a normal champion select, this is normally acceptable. Against someone like Darius, it's not. He destroys at lower Elo's because of this because his job is punishing mistakes, which is considered toxic. It's not fun learning how much you can trade when Darius will out trade you every time, and then ult you for the kill.
Champions that are built around Control (Heimerdinger, Teemo) They have mechanics which stop positive mechanics (Trading, Ganking other lanes, etc). Walking into a bush and dieing to a teemoshroom is not "Damn he outplayed me". It's frustrating and sad, feelings that shouldn't be in a game about co-operation and teamwork. There scared to buff Heimerdinger because he has a lot of aspects that prevent fun game-play. He currently doesn't need to be in lane to CS, his damage is locked up in stationary places,he has a hard time "Following up" on any plays your team makes because of this.
Champions that troll your own team: (Jarven, Anivia, Thresh, blitzcrank etc)These are champions you'll often see "Teh PLayz OMGAWDLOLOL" with. There toxicity comes from when they.. well make mistakes, or troll you. Ever been that mid traped in malzahar's pool because YOUR jarven ganked the lane? I know I have.. running away in a jungle fight Anivia walls for protection! Trapping you from your only escape route. R.I.P Summoner 1. Mechanics that can be misused against your own team are considered toxic.
Champions with few counters(Yi): Yi has been considered Toxic because or his utility and massive damage. Even when yi is doing poorly, he will be able to backdoor effectively, an assassinate squishy targets. He has a lot of "Get out of jail free" cards, and if used effectively is very difficult to deal with unless specifically countered. This would be the "OmGAWD OP" Toxicity. It's not one part of the kit that is overpowered, but the synergy between each skill allowing them to deal with almost any situation. Vi, Xin zhao, Jax, Jayce are champions that have been in this sphere but mostly have been balanced. Yi was recently here after his rework.
as a TL;DR or if I messed up with my English: Champions that remove fun aspects from the game, or champions that you feel powerless against are considered toxic. The game should be as fun as possible. Whether you are winning or losing, professional or learning, you should be enjoying yourself. Champions that take away from this enjoyment as a matter of mechanics are Toxic because they spoil this fun, which degrades the very purpose of the game.
