There are some known "skins-lore".
Super Galaxy Rumble is, I think, Rumble after he tried to protect Bandle City but didn't manage to, so he upgraded himself and Tristy (the machine).
Blackfrost Anivia is Anivia in a future where Lissandra takes over Valoran and everything was turned into ice/Black Ice.
The Battlecast skins line is for champs who joined (by will or were forced to) Viktor's Glorious Evolution. And Creator Viktor is how he's armored as the Creator and Leader of the Battlecast Army and the Glorious Evolution.
Bloodmoon Kalista is how Ionians see and imagine Kalista, as the legends says and the tales tell, and Bloodmoon Shen and Akali are them in the Ionian Bloodmoon Festival (which can be read about in the Wiki/JoJ). And I assume that Bloodmoon Thresh and Elise are like with Kalista- How Ionians imagine and see them after all the legends and tales that have been told about them.
Secret Weapon Zac is Zac in a time in which his "parents" never stole him from the lab, so he was raised by the scientists.
Redeemed Riven is Riven, well, redeemed. Exiled Morgana is, too, literately what it is, if you read her lore.
I think that Divine Soraka is how she looked like before she hurt anyone, if I remember correctly.
The Dragon Slayers is what it is. It was also stated in Feats of Braum that Braum killed a dragon once and a saved a village from it.
Pentakill is one of the most famous bands (if not THE most famous) and can be read about more here.
The Heist was an event in which Twitch and Evelynn teamed up in order to steal from (Piltover's?) the Legacy Vault. So the Officers Caitlyn and Vi needed backup, so they sent Captain Volibear and Constable Trundle to help and try to catch the two thieves.
I don't know for sure, but I read somewhere that Spirit Guard Udyr is Udyr in a time where his master haven't died (I think) and he is still protecting the (his master's?) temple. I haven't read Udyr's lore completely so if someone knows really what's the story behind SGU, I'd be glad to know. :)
Pulsefire Ezreal may be Ezreal, after joining Jayce's Resistance? More Armored with Pulsefire tech, but I'm not sure about this either.. :P
I read that Hextech skins are supposed to be part of Jayce's Resistance, but I don't know..
The Mecha skins seem to be Rumble's, Heimerdinger's, and Zigg's Mechanical Combat Units. But they seem to be more of a reference to Pacific Rim, simply. :P
There are probably a lot more I am missing, but that's what I remember. :D