You rank your account up, not the individual characters.
Do you keep your levels after every match?
Well, you do, in-game.
Each game, your character goes to level 1, and goes up to level 18, earning your 4 skills in any way you choose/can.
This, but being higher Summoner level does do something. There are mastery trees, and you get 1 mastery point per level. So a new Summoner has 1 mastery and Level 30 Summoners have 30. 30 is max Summoner level. Matchmade games will match you close to your Summoner level.