So I paired a song with every champ in League. The vast majority are video game or cartoon songs. With some I picked the song because it fits the champ or the champs playstyle and with others because it parodies the champ. Enjoy. I put a link for the songs that may be too obscure to find just by searching.
Aatrox- paradise lost of blood of another
Ahri- MoF Extra Stage Boss - Suwako Moriya's Theme - Native Faith
Akali- Okamiden- Kurow Loses Faith
Alistar-
I am cow
Amumu- Sad violin
Anivia- Laverre City - Pokémon X & Y Music Extended
Annie- Gruntilda's lair theme
Ashe- Death or Sovngarde
Azir- Soundcritters - Unity
Blitzcrank- Beep, Boop, Everyone
Brand- The Roof is on Fire
Braum- Katyusha
Cailtyn- Team Fortress 2 Music - Sniper's Theme
Cassiopeia- We love burning town
Cho'gath- The godzilla 1998 song "come with me"
Corki- British Grenadiers
Darius- Doom 3 - Main Theme
Diana- Fate/Stay Night OST - Tenchi Hou Take
Dr. Mundo- The Blue Danube Waltz
Draven- Elton John - Ego
Elise- Fur Elise
Evelynn- Pink Panther
Ezreal- Haru Haru Yukai
Fiddlesticks- Spooky scary skeletons
Fiora- La Marseillaise
Fizz- Surf Theme Extended - Pokemon Omega Ruby & Alpha Sapphire
Galio- Gargoyles Theme
Gangplank- He's a Pirate
Garen- Spin to win
Gnar- Donkey Kong Country: Tropical Freeze Soundtrack - Homecoming Hijinx ~ Snowmads' Island Theme
Gragas- Charlie Mopps - Beer Beer Beer Song
Graves- The good the bad and the ugly- main theme
Hecarim- The Pulse of the Dead
Heimerdinger- Erectin' a river
Irelia- One final effort- Halo 3
Janna- Song of storms
Jarvan IV- Tsuna theme
Jax- Rocky Theme
Jayce- Wham's theme Jojo's Bizzare adventure
Jinx- GET JINXED!
Kallista- HUNTRESS - Zenith
Karma- Karma Chameleon
Karthus- What up People Maximum the hormone
Kassadin- Phantom Lord
Katarina- Viral Theme
Kayle- The Legend of Zelda 25th Anniversary Symphony: Ballad of the Goddess
Kennen- Thundercloud Okamiden
Kha'zix- Zerg theme
Kog'maw- Gnaw theme
Leblanc- Camille Saint-Saëns - Danse Macabre
Lee sin- Wong Fei-Hong - Once Upon A Time In China Theme
Leona- Two Steps From Hell - Protectors of the Earth
Lissandra - Vivaldi- Four Seasons (Winter)
Lucian- Super Smash Bros. 3DS - Jogging Theme / Running Theme (Punch-Out!)
Lulu- Saria's song
Lux- Sunshine, Lollipops And Rainbows
Malphite- We will rock you
Malzahar- Shift al mani
Maokai- 25 Smaug
Master yi- The sun rises okami
Ms Fortune- sea shanties - spanish ladies
Mordekaiser- Pulse of the Dead
Morgana-Shadowman ashes
Nami- Zora's Domain
Nasus- Walk like an egyptian
Nautilus-Aboard the GFS Valhalla
Nidalee- Roar Katy Perry
Nocturne-Alundra Nightmare
Nunu- Icirrus City - Pokémon Black & White Music Extended
Olaf- Awesomenauts - Skolldir Killing Spree
Orianna-
Swan Lake Techno remix
Pantheon- THIS IS SPARTA!
Poppy- Shadow of the Colossus OST - Revived Power
Quinn- Birds of Prey
Rammus- Wrecking Ball
Rek'sai- Jaws theme
Renekton- Schanppi Das Kleine Krokodil
Rengar- Predator theme music
Riven- Zinnia theme
Rumble- Trombe
Ryze- mc chris - i'm a fuckin wizard
Sejuani- Tales of Graces - Battle Theme (Ice)
Shaco- Psychonauts OST - The Meat Circus
Shen- Tea Song of the Xiang River
Shyvanna- The Dragon and the Wolf (A song of Ice and Fire)
Singed- The Stanley Parable - Following Stanley
Sion- Exorcist - From Ao No Exorcist
Sivir- Final Fantasy VII OST - Fight On!
Skarner- Lanayru Mining Facility (Past) - The Legend of Zelda: Skyward Sword Music Extended
Sona- Rondo alla Turca, Mozart
Soraka- [Persona 4 OST] 05 - New Days
Swain- Mother 3 - High Highness Remix
Syndra- Vs. Ghetsis - Pokémon Black & White Music Extended
Talon- Metal Gear Solid 4: Main Theme
Taric- Abba - Dancing Queen
Teemo- Maple leaf rag
Thresh- Death Metal: Hell on Bare Feet
Tristana- Piotr Ilich Tchaikovsky - 1812 Overture (Finale)
Trundle- Troll Song
Tryndamere- Battle For Kattegat
Twisted Fated- Phil Bellante - All In
Twitch- Pyrite Town - Pokémon Colosseum Music Extended
Udyr- Street Fighter IV Ryu's theme
Urgot- Neil Young - Piece Of Crap
Varus- Arrow Theme
Vayne- clubbed to death - Matrix
Veigar- Carl Orff - O Fortuna ~ Carmina Burana
Vel'Koz- Trance/Techno - Evil Angel
Vi- Here comes Vi
Viktor- Daft punk teechnology
Vladimir- Hellsing Opening (Full Song)
Volibear- Bear nessecities
Warwick- Skillet - Monster
Wukong- Donkey kong country returns main theme
Xerath- The Impossible Game Soundtrack-Level 1 Fire Aura
Xin Zhao- Be a man- Mulan
Yasuo- Okami - Ryoshima Coast
Yorick- Stone Tower Temple
Zac- Awesomenauts - Scoop's Killing Spree
Zed- Shadow - The Arrival at the last Quarter
Ziggs- Bomberman Land Wii OST - Main theme
Zilean- Time Warp
Zyra- Music of Cataclysm - Night Elves
Aatrox- paradise lost of blood of another
Ahri- MoF Extra Stage Boss - Suwako Moriya's Theme - Native Faith
Akali- Okamiden- Kurow Loses Faith
Alistar-
Amumu- Sad violin
Anivia- Laverre City - Pokémon X & Y Music Extended
Annie- Gruntilda's lair theme
Ashe- Death or Sovngarde
Azir- Soundcritters - Unity
Blitzcrank- Beep, Boop, Everyone
Brand- The Roof is on Fire
Braum- Katyusha
Cailtyn- Team Fortress 2 Music - Sniper's Theme
Cassiopeia- We love burning town
Cho'gath- The godzilla 1998 song "come with me"
Corki- British Grenadiers
Darius- Doom 3 - Main Theme
Diana- Fate/Stay Night OST - Tenchi Hou Take
Dr. Mundo- The Blue Danube Waltz
Draven- Elton John - Ego
Elise- Fur Elise
Evelynn- Pink Panther
Ezreal- Haru Haru Yukai
Fiddlesticks- Spooky scary skeletons
Fiora- La Marseillaise
Fizz- Surf Theme Extended - Pokemon Omega Ruby & Alpha Sapphire
Galio- Gargoyles Theme
Gangplank- He's a Pirate
Garen- Spin to win
Gnar- Donkey Kong Country: Tropical Freeze Soundtrack - Homecoming Hijinx ~ Snowmads' Island Theme
Gragas- Charlie Mopps - Beer Beer Beer Song
Graves- The good the bad and the ugly- main theme
Hecarim- The Pulse of the Dead
Heimerdinger- Erectin' a river
Irelia- One final effort- Halo 3
Janna- Song of storms
Jarvan IV- Tsuna theme
Jax- Rocky Theme
Jayce- Wham's theme Jojo's Bizzare adventure
Jinx- GET JINXED!
Kallista- HUNTRESS - Zenith
Karma- Karma Chameleon
Karthus- What up People Maximum the hormone
Kassadin- Phantom Lord
Katarina- Viral Theme
Kayle- The Legend of Zelda 25th Anniversary Symphony: Ballad of the Goddess
Kennen- Thundercloud Okamiden
Kha'zix- Zerg theme
Kog'maw- Gnaw theme
Leblanc- Camille Saint-Saëns - Danse Macabre
Lee sin- Wong Fei-Hong - Once Upon A Time In China Theme
Leona- Two Steps From Hell - Protectors of the Earth
Lissandra - Vivaldi- Four Seasons (Winter)
Lucian- Super Smash Bros. 3DS - Jogging Theme / Running Theme (Punch-Out!)
Lulu- Saria's song
Lux- Sunshine, Lollipops And Rainbows
Malphite- We will rock you
Malzahar- Shift al mani
Maokai- 25 Smaug
Master yi- The sun rises okami
Ms Fortune- sea shanties - spanish ladies
Mordekaiser- Pulse of the Dead
Morgana-Shadowman ashes
Nami- Zora's Domain
Nasus- Walk like an egyptian
Nautilus-Aboard the GFS Valhalla
Nidalee- Roar Katy Perry
Nocturne-Alundra Nightmare
Nunu- Icirrus City - Pokémon Black & White Music Extended
Olaf- Awesomenauts - Skolldir Killing Spree
Orianna-
Pantheon- THIS IS SPARTA!
Poppy- Shadow of the Colossus OST - Revived Power
Quinn- Birds of Prey
Rammus- Wrecking Ball
Rek'sai- Jaws theme
Renekton- Schanppi Das Kleine Krokodil
Rengar- Predator theme music
Riven- Zinnia theme
Rumble- Trombe
Ryze- mc chris - i'm a fuckin wizard
Sejuani- Tales of Graces - Battle Theme (Ice)
Shaco- Psychonauts OST - The Meat Circus
Shen- Tea Song of the Xiang River
Shyvanna- The Dragon and the Wolf (A song of Ice and Fire)
Singed- The Stanley Parable - Following Stanley
Sion- Exorcist - From Ao No Exorcist
Sivir- Final Fantasy VII OST - Fight On!
Skarner- Lanayru Mining Facility (Past) - The Legend of Zelda: Skyward Sword Music Extended
Sona- Rondo alla Turca, Mozart
Soraka- [Persona 4 OST] 05 - New Days
Swain- Mother 3 - High Highness Remix
Syndra- Vs. Ghetsis - Pokémon Black & White Music Extended
Talon- Metal Gear Solid 4: Main Theme
Taric- Abba - Dancing Queen
Teemo- Maple leaf rag
Thresh- Death Metal: Hell on Bare Feet
Tristana- Piotr Ilich Tchaikovsky - 1812 Overture (Finale)
Trundle- Troll Song
Tryndamere- Battle For Kattegat
Twisted Fated- Phil Bellante - All In
Twitch- Pyrite Town - Pokémon Colosseum Music Extended
Udyr- Street Fighter IV Ryu's theme
Urgot- Neil Young - Piece Of Crap
Varus- Arrow Theme
Vayne- clubbed to death - Matrix
Veigar- Carl Orff - O Fortuna ~ Carmina Burana
Vel'Koz- Trance/Techno - Evil Angel
Vi- Here comes Vi
Viktor- Daft punk teechnology
Vladimir- Hellsing Opening (Full Song)
Volibear- Bear nessecities
Warwick- Skillet - Monster
Wukong- Donkey kong country returns main theme
Xerath- The Impossible Game Soundtrack-Level 1 Fire Aura
Xin Zhao- Be a man- Mulan
Yasuo- Okami - Ryoshima Coast
Yorick- Stone Tower Temple
Zac- Awesomenauts - Scoop's Killing Spree
Zed- Shadow - The Arrival at the last Quarter
Ziggs- Bomberman Land Wii OST - Main theme
Zilean- Time Warp
Zyra- Music of Cataclysm - Night Elves