First of all, it's good that you came here now, rather than the day after getting a permanent suspension and asking us for tips on trying to overturn it.
Secondly, if it's been for behavior (things like flaming) then the easiest thing to do is mute everyone. In champ select, you can tell your team that you will be muting chat, but will be communicating with pings, and ask them to do the same if they are needing your help, wanting to gank, etc. Basically, if you never see anything negative said by someone, you're less likely to get defensive and say something back to them.
Third, that's only half the flame war down. The other half of it is going to involve good, old fashioned self-control. You've got to make yourself not say anything to them first. Some people do it by unbinding their enter key. Some people hide the chat window to where it would take too long for them to get it back to type into it. Whatever it takes for you to keep from lashing out at them, do it. Because at this point, you cannot say anything to them any more.
The good news is, you don't stay on this last punishment tier forever. After a period of time of good behavior, you will drop down to where you aren't in danger of an instant permaban on your next offense. There is no set time for this, it's all based on the amount of games you play, and more importantly, showing reform in those games. This means not getting yourself reported. But, something else to keep in mind, there are trolls in League who just report people for no reason. If you are reported, but haven't done anything... you won't be punished. It has to be viable reports for you to be punished. If they report you for hate speech, but you didn't type in chat, that report will not do anything.
One last thing to keep in mind... other sorts of reports, for things like trolling, or intentionally feeding, can count against you in this situation. So, no building 6 Tears of the Goddess on Riven, or 6 Boots on Elise, don't sell all your items in the last minute of a game and buy something dumb. Just play the game, have fun, and be respectful to the people you get stuck with. You only have to deal with them for 30 minutes. They are not worth losing your account. Ignore them, report them if you have to, and move on.