To Blitzcrank: "The true inventor knows his invention best. Your defeat was imminent."
To Heimerdinger: "Your toys are were no match for my superior crafts!"
To Ziggs: "Your explosives are too erratic and fragile. You lack focus."
To Rumble: "Your over-reliance on a third-rate armored battle construct was your undoing."
To Singed: "Who in their right mind would even imagine that steel was susceptible to poison?"
To Warwick: "Truly, your great mind has been tarnished by your irrational bloodlust."
To Mundo: "Pain means nothing to the Machine Herald."
To Viktor: "The glorious evolution will be realized. I will see to it, myself."

Varus --> Riven "Once a dog, always a dog!"
Darius --> Draven "Nothing personal, Draven. You just need to become stronger."
Draven --> Darius "Not so little now, am I?"
Karthus --> Yorick "Death...is beneath me..."
Yorick --> Karthus "Your death was long overdue, lich king."
Katarina --> Cassiopeia "You and the rest of our sisters were always my inferiors."
Cassiopeia --> Katarina "There's more than one way to skin a Kat."
Wukong --> Cho'Gath "If you're the strongest that the Void has to offer, I've already beaten you a thousand times over."
Cho'Gath --> Wukong "Too much hair, not enough meat, but I suppose this will have to do..."
Xerath --> Ryze "Foolishness, mage. Foolishness."
Ryze --> Xerath "Never come back, you arcane abomination!"

Here are some quotes of champions killing themselves...

Ahri: "A shame to ruin such a pretty face. Good thing it wasn't mine. (giggle)"
Akali: "Impostor eliminated."
Alistair: "Nice try, my friend, but I'm the best!"
Amumu: "Wait. Was that...me?!"
Blitzcrank: "YOU ARE AN EMBARRASSMENT TO GOLEMS EVERYWHERE."
Dr. Mundo: "Is Mundo really that ugly?!"
Fizz: "Oh, my--what have I done?! I'm all alone again!"
Gragas: "Hey. This fellow looks just like me. (laugh)"
Heimerdinger: "Mental note: fix bugs in cloning machine."
Jarvan IV: "Jarvan Lightshield IV shares his name with no one!"
Kayle: "Farewell, impostor!"
LeBlanc: "I think I need to work on my illusions more..."
Morgana: "I detest only one type of being more than a tyrant: an impostor!"
Nasus: "It appears I must hone myself more..."
Olaf: "I am the better man!"
Renekton: "I will never stop killing you, impostor!"
Swain: "Somehow, Evaine must be behind this..."
Teemo: "Next time, don't disguise yourself as me. You'll only make it worse for yourself."
Tristana: "I guess that makes me the real me."
Trundle: "One troll to another, pal: GIVE UP!"
Varus: "You have no right to live, much less impersonate me!"
Xerath: "Only one may transcend the boundaries of magic today."
Zilean: "Two of us cannot exist at the same time."

Not much but, here I go.

Shen --> Rammus: "Words don't hurt more than steel."
Shen --> Akali: "You still need training, Fists of Shadow."
Shen --> Kennen*: "It'll hurt more if you stay still and rapidly pressing the LP button."
Shen --> Galio: "I destroy stones for training."
Shen --> Ahri: "An iron-will resists charms."
Shen --> Jax: "Try training with your head next time."
Shen --> Blitzcrank: "There's the shutdown button."
Shen --> Viktor: "Changing your body doesn't change anything."
Shen --> Teemo: "Pee-ka-boo."
Shen --> Varus: "Vengeance has no purpose."
Shen --> Riven: "Funny, you're keeping the bunny suit?"
Shen --> Talon**: "Take the leap of faith, and stay hidden."
Shen --> Twisted Fate: "Here's a magic trick. Where's your cards?"
Shen --> Annie: "I don't fight with kids."
Shen --> Shen: "Canceling: Clone Technique."

((* - Street Fighter Blanka Reference I ** - Assassin Creed Reference))

More quotes for champions killing themselves...

Anivia: "So alike, yet so different..."
Brand: "Leave the fire play to the experts, fool."
Corki: "Bogey dow--wait a sec. I think that was me!"
Galio: "And now, to thank your creator..."
Graves: "This map ain't big enough for the two of us!"
Lee Sin: "Assuming my form will give you my power, but not my skill."
Lulu: "Next time, I'll dress up as you instead."
Lux: "Yeah. This isn't weird or anything..."
Maokai: "Another tree?! How dare those summoners turn me on one of my own!"
Nocturne: "To die by one's own hands. Truly the greatest nightmare of all."
Orianna: "Maybe this is why I am an only child..."
Riven: "Sometimes, I wonder why that didn't happen to me sooner."
Shyvana: "I knew when I first laid eyes on you that you were no dragon."
Twisted Fate: "Looks like I had all the aces that time."
Vladimir: "So much blood... I love it."
Wukong: "You can always tell a monkey, but you can't tell him much."

Ah man, if I had the time to go through page by page and thumbs up everyone, I would. I so hope Riot does this! I mean, they don't have to do every champion against every champion (it would be nice, but there are a lot of champions and more on the way, it would get pretty difficult to keep up). At least the rivals and selves would be nice. Just the rivals or themselves would be a nice start. I would put in one of my own, but it probably wouldn't be good, so I guess you guys will just have to settle for my congrats =).

EDIT:
Wait, heatman gave me an idea. WW VS Nasus should have an epic line either way, it's the ancient dog-wolf feud! Now... I will post again when I think of something awesome.

Even more quotes for champions killing themselves...

Darius: "Thought you could copy me, huh? Don't make me laugh."
Draven: "Sorry, but Draven doesn't share the spotlight with anybody."
Evelynn: "You call that assassination?"
Fiddlesticks: "So that's what the inside of my head looks like..."
Gangplank: "Say hello to me pappy for me, whalebait!"
Garen: "There are many warriors, but only one Garen Crownguard!"
Leona: "It seems the sun wasn't with you today."
Malphite: "Sorry."
Malzahar: "Only one prophet is needed here today: me."
Sejuani: "Only I deserve to rule Freljord!"
Vayne: "Another illusion..."
Volibear: "A true Ursine has no need to hide his name."
Xerath: "Your pale imitation is as nothing before my might!"
Xin Zhao: "The Lightshield dynasty has need of only one guardian."

Ezreal-> Cho Gath with the Jurassic skin "You belong in a museum!"