You get chests from either you or someone in a premade getting an S- or above. But, you can only get one chest per champion a season.
So like if you're playing Syndra and your friend that you queued up with gets an S you will get a chest. But, if you get an S on Syndra you won't get another chest on her because you already got one for the season.
Key frags are dropped by winning games and that's it. They reset every month (the drop rate) so someone who plays 1 game a day will actually end up getting the same number of key fragments as someone who plays 10 games a day.