This has been done before certainly, but we should get a definitive list going.
Pick a song that you think showcases each champion's personality/character the best. For example:
https://www.youtube.com/watch?v=nuLsxo20Rmo
https://www.youtube.com/watch?v=7AensT8ntoY
https://www.youtube.com/watch?v=9LjbMVXj0F8
https://www.youtube.com/watch?v=6sL6WrWHkhw
https://www.youtube.com/watch?v=_LBmUwi6mEo
Let's see what you can come up with!
ummm... oh to hell with it:
https://www.youtube.com/watch?v=CEEPaYD5KZE
5 points who saw this coming:
https://www.youtube.com/watch?v=ET3w3vzQW6g
https://www.youtube.com/watch?v=06WyJfUH5E8
https://www.youtube.com/watch?v=QEjgPh4SEmU
...I regret nothing... (I could not find a metal version for the life of me... which would have worked better...)
https://www.youtube.com/watch?v=m86nMHfvl7s
https://www.youtube.com/watch?v=_Vj092UgKwQ
Might not be perfect, but I think this emulates 
fairly well.
https://www.youtube.com/watch?v=TRzPN_X1fqE
https://www.youtube.com/watch?v=FmuFBK26TtE
I don't why but to me it just works.
https://www.youtube.com/watch?v=CKh0dLIuIu8
You were thinking too
https://www.youtube.com/watch?v=AVy7YPNP_zI
because Olaf
https://www.youtube.com/watch?v=monN9Ok0El8
It Feels like Lucian's Style
https://www.youtube.com/watch?v=Nba3Tr_GLZU
This is what people feel when Nocturne is hunting them
Oh no...here we go. This is gonna be a wall of fucking text. I really like music. Don't judge me.
https://www.youtube.com/watch?v=wKe-oqJVgZI Sung this to Lux.
https://www.youtube.com/watch?v=8XVtfWxNMss Primus sucks.
https://www.youtube.com/watch?v=DXafRKe9klA Suitable for a tale of mistakes and redemption.
https://www.youtube.com/watch?v=R2F_hGwD26g
https://www.youtube.com/watch?v=OUGbu6Jcrmw He "lives" in a sense that he grows stronger from others dying. Ya dig?
(He is tied in with Ionia now) 
Ionia in general. I would think they could stop in fighting long enough to fend off threats from outside. https://www.youtube.com/watch?v=Yo5fE-Zl-V4
https://www.youtube.com/watch?v=XZp3Mtn-YsI
https://www.youtube.com/watch?v=vefADDxiyeM I feel like Vi has a MUCH greater understand of what street life is like for children having been raised by a street gang. Having met Cait she found her way out, but she'll never forget what it was like. Maybe not as energetic as her actual theme, but not every part of a character is.
https://www.youtube.com/watch?v=RZul_mSFczs Just the right amount of "fuck the man".
https://www.youtube.com/watch?v=J8ExPHzQqcs Im not really even sure why. Think more the music itself. Kind of bouncy and silly, but still a tad serious. I dont know what it is about Rumble that just screams Primus at me. I blame Les.
vs 
https://www.youtube.com/watch?v=C9hR5Byl8Ys
https://www.youtube.com/watch?v=johEASsTUNw Kind of obvious. I miss old ID software.
https://www.youtube.com/watch?v=ZreE2lZBR5o
https://www.youtube.com/watch?v=cwqxLW9138g
https://www.youtube.com/watch?v=iS0Zxctzrk0
https://www.youtube.com/watch?v=FfbKrphheQk
The city of Piltover https://www.youtube.com/watch?v=4V4IEV8l-gA
I'm sure ill chew on this forever and come back with like 20 more. Music is important yo.
