Ok so we all know that custom skins appear to not be working, since this has let down many leaguecrafters, skinners, mod-ers, and the classic members alike there are some things that should be pointed out.
There already are two workarounds (credits to ItzWarty for the original tweak and lordgreggreg/LGG for the new installer) that both appear to be functional, and you can find two link of them in this thread's first post.
Note that those workarounds appear to not be functional for everybody, but dont let yourselves down, it has been only three days that the US had the new launcher and just one since the EU release, that means that the workarounds are in a quite "basic" state and are most likely to get better and more functional over time.
For the direct links.
ItzWarty's solution(Takes more time, but appears to be functional for everyone)
lordgreggreg's/LGG's installer (Faster, easyer and cleaner to use, but still in an early state, bugs might appear)
Also ItzWarty realsed his new RAF Manager, so check it out
So dont let yourselfs down, the League of Legends community always proved to find ways to achieve its goals, and will once more do it!
Garona and the Moderating crew.
*NOTE : Some skins might not work with the installer, if you have any problems, please contact LGG in the thread linked.