Here are the teams and their names.

Droids: Urgot, Viktor, Orianna, Blitzcrank, Rumble
Ice Ice Baby: Sejauni, Janna, Ashe, Tryndamere, Olaf
Triforce: Akali, Kennen, Shen
The Magicians: Shaco, Karma, LeBlanc, Twisted Fate
Animaux: Swain, Cassiopeia, Shyvana, Udyr
Foresters: Nidalee, Maokai, Malphite
Pentakill: Sona, Karthus, Mordekaiser, Yorick
Yordles: Poppy, Teemo, Tristana, Amumu
2Stronk: Volibear, Alistar, Sion, Dr. Mundo, Kayle
Hot stuff: Evelynn, Morgana, Caitlyn, Ahri, Katarina

So I'm going to write a story about a war in Runeterra. By using in-game quotes.

Everything has fallen apart and Rumble left the yordles to join Urgot, Viktor, Blitzcrank and Orianna.

Urgot: There's no escape. We have to launch an attack.
Viktor: All will soon change for the better.
Orianna: Motivating.
Blitzcrank: The time of man has come to an end.
Orianna: We are as one.
Rumble: Let's get in the fight!
Urgot: They will know fear.
Orianna: We will kill your enemies. That will be fun.

Meanwhile, Sejauni, Ashe, Janna, Tryndamere and Olaf were hiding in Freljord. Janna was recovering from her duel with Katarina.

Tryndamere: This'll be a slaughter.
Ashe: Agreed.
Sejauni: This is our fight!
Olaf: Finally some fun.
Sejauni: Victory at any cost.

Shen, Akali and Kennen were inseparable. As always.

Kennen: Their final moments approach.
Shen: From the shadows.
Akali: Whatever's necessary. We have to win this.
Kennen: For Ionia.

The Magicians and Animaux had started a fight, and spells were flying all over the ruins of Noxus. Karma used her fans to shield her from Cassiopeia's venom, LeBlanc dodged Shyvana's attacks by distortioning away, Shaco was laughing at Swain and Udyr was charging at Twisted Fate.

Shaco: Just a little bit closer!
Swain: If you insist.
Shaco: Look... behind you.
And with a laugh, he stabbed Swain in the back.
Shaco: The joke's on you!

Cassiopeia: Surrender to me.
Karma: You should feel honored to face me.
Cassiopeia: I strike.
Karma: Guess what's about to hit the fan?
Cassiopeia used her twin bite, but only hit Karma's fan. With her other fan, she killed off Cassiopeia.
Karma: Negotations have ended.

Shyvana was looking around for LeBlanc, then she heard a noise behind her.
LeBlanc: Surprised to see me?
Shyvana: Looks like the quiet before battle... just ended.
LeBlanc: Looks can be deceiving.
Shyvana: Stop that!
LeBlanc: Patience, Shyvana.
Shyvana transformed into a dragon.
Shyvana: Behold my true form.
LeBlanc: Tricky, aren't you?
Shyvana: Fear me.
She charged at LeBlanc, who disappeared.
LeBlanc: It's all smoke and mirrors.
Shyvana: By the blood of my father, I will end you.
LeBlanc laughed and threw a chain at Shyvana. Then she cloned herself and her clone mimiced her. Shyvana fell with a roar.
LeBlanc: For a moment I thought I'd broken a sweat.

Udyr roared at Twisted Fate, who was throwing cards at him from everywhere. When he ran towards him, he was gone.
Udyr: Stay where you are!
Twisted Fate: I'd rather not.
He threw a card that hit Udyr in the knee.
Twisted Fate: It's my lucky day.
Udyr: Weapons are for the weak...
Twisted Fate: Since when are cards weapons?
Udyr: Quiet! Your death will...
But Twisted Fate had hit his neck with a card, and Udyr dropped, his head rolled to Shaco's feet.
Twisted Fate: The house always wins.