DisBackForMore,
There's been a lot of good advice in this thread so far, so I really encourage you to hear it out. By all means, ignore the haters. It's okay to get angry sometimes, anyone that pretends otherwise is bullshitting you. It happens, we're human.
People who will mock you for it on these boards are not worth responding to.
To your particular issue at hand. If your comfortable, I'd encourage you to link or copy and paste the text of the chat that was sent to you for your ban. No pressure if you don't want to, but it does help others better understand where your behavior may be going wrong.
In the heat of the moment we may feel like we're justified in our actions. In our minds, it is Us Vs. the World. But other people can help us understand perspective a little bit better, and by sharing the chat you may gain insight in to your own actions.
I'm going to make an assumption in your case based off the post you made. If it's wrong; forgive me; I'm happy to adjust it based off the actual chat. (I don't have access to my tools right now so I can't view it)
You mention that other players were saying terrible things to you and that it angered you. In response, you asked them to stop. If you ended up suspended, I'm going to guess you said some mean things in response as well.
First, it's important to know that if what you said about the other players' comments is true, then that's un-excusable. Telling someone or their family members to die is NOT okay. That behavior is the kind of stuff our disciplinary systems are looking out for. You. I and everyone else in League shouldn't have to deal with that, so it's important that we report it after the end of the game when we see it.
But if you respond to that level of toxicity from others with your own toxicity, then you're not helping the situation at hand. You're not helping yourself, you're not helping your team. You're not going to get the other people to stop (You'll probably encourage them by feeding them information).
If anything, you're likely to hurt yourself and your team. Let's take a hypothetical game where you are playing Ranked Queue on Summoner's Rift. There are five players on the team, including you.
You
Player 2
Player 3
Player 4
Player 5
10 minutes in, Player 2 says really nasty things to you.
15 minutes in Player 5 says really nasty things to you.
You snap. You've been playing 15 minutes of this game, trying your hardest and now you have Player 2 and Player 5 ragging so hard on you; just tearing you a part with every insult in the book.
In retaliation you unleash a stream of your own attacks. The kind of stuff that will make your grandmother blush.
What's the end result here? At 15 minutes there were 2 toxic players in the game and 3 non-toxic players in the game. Now you've joined in, and there are 3 toxic players in the game.
Perhaps three minutes later Player 4 decides why the hell not, and they start being a jerk too. Now there are 4 toxic players.
We don't want that. You don't want that. The other players in the game don't want that.
In an alternate scenario let's say instead of unleashing on the two toxic players you muted them. There are no more insults coming from them. They're likely to give up trying to insult you (since you're not giving them the attention they want) and if they don't, then who cares! you wouldn't even know they're saying anything since they are muted. All that wasted effort on your part for them to troll you. :P
So now we have a situation where you ignored them and stopped the toxicity where it is. Two toxic players, three non-toxic players. You have the mental energy to focus on the game without their terrible comments. The other two players in the game don't have to worry about another person becoming toxic either. Best case scenario, the two toxic players get tired of their insults and start playing.
If those two toxic players don't? Well, let's be honest; that sucks. BUT now you have the trump card. You report them at the end of the game. The disciplinary systems take a look and ban them.
If they report you in retaliation? Who cares! You didn't say anything to them, so their report is disregarded.
At the end of the day; they're banned. You're not.
This is what I'd encourage you to do the next time. In the situation that led to your ban, you saw yourself a victim; BUT, you were also an aggressor by adding your own toxicity to the game, ruining it for everyone else. In the future, you can make sure you're not either by muting/ignoring toxic players and reporting them at the end.