PSA: Before Making An "I Was Banned for Toxic Behavior" Thread
Consider this: You were banned for being toxic, yes? Then is going to the boards and making an angry thread blaming everyone else really going to solve anything? You're just continuing the same behavior that landed you that ban in the first place.
Read your own chat logs, then pretend you're a stranger having those things said to you. You might just see why this behavior isn't at all productive and just ends up feeding itself.
If you think you can appeal your ban or get it overturned by being aggressive, then you're missing the entire point of the ban. Cool it, and think about it for a moment. If you REALLY wanted to get something done about your ban, the first thing you should be doing is perhaps showing that you can behave in a civilized manner. If more people did this, then maybe Riot would actually consider offering ways to repeal the bans. (If you've heard of the Level 20 Challenge, then yes, Riot HAS considered such things in the past.)
The fact is, as long as people continue to not learn anything from their bans, nothing more will be done with them. If you really want to fight for your account, do so in a way that shows that you actually DESERVE that second chance. I'm not saying you're guaranteed to get it, though. Frankly, the chances of it ARE slim, but that slim chance is still better than the no chance that most leave themselves with.
I've seen too many ban threads all end the exact same way, and for the exact same reasons. It doesn't NEED to be that way if you don't let it.
Edit: Some other helpful/informative threads to have a look at from other banned players:
http://boards.na.leagueoflegends.com/en/c/player-behavior-moderation/L76T1Taa-from-a-banned-player-thinking-about-your-actions (this one's my own thread, but it tells more about my own ban and the aftermath)
http://boards.na.leagueoflegends.com/en/c/player-behavior-moderation/HE8cdhGE-anger-and-regrets
http://boards.na.leagueoflegends.com/en/c/player-behavior-moderation/41pnKA5A-permabanned