Is it Illegal to use custom skins while streaming Twitch in League?

Actually, unless something has changed in the recent past, there isn't anything explicitly forbidding the use of custom skins. However, certain programs that provide custom skins also contain applications as a part of the software that are banned, so using them, even if you don't use the other functions WILL get you banned. If you look at this thread by Wuks, it details a list of third party programs and Riot's stance on them fairly clearly. It isn't complete/set in stone, as new things are constantly being developed/changed, but Wuks does a great job of adding to this over time, so kudos to him!

http://boards.na.leagueoflegends.com/en/c/player-behavior-moderation/5dmkJinG-3rd-party-applicationsmodifications-whats-allowed-and-what-isnt

The skins themselves aren't forbidden, provided that they are not meant to be copies of skins that Riot currently has for sale. That's definitely a no-no. But I can think of a few streamers who have used custom skins on Twitch, or on league related videos, one of which was the top rated streamer on Twitch for a long time.

That being said... any modification of the client is a risky business, and while the use of the skins isn't necessarily forbidden at this time, that doesn't mean that it may not become forbidden in the future. Not to mention, it can be very tricky finding out which of the softwares that offer the custom skins also come with the banned applications.

EDIT: custom skins are not illlegal. It's illegal to use custom skins, period. You could get permanently banned for messing with in-game files. Check out this thread to see what third party programs are allowed.

Some relevant red posts on this http://boards.na.leagueoflegends.com/en/c/miscellaneous/5kx4F7VE-reminder-third-party-applications http://boards.na.leagueoflegends.com/en/c/riot-official/EbE4QMZW-an-update-on-3rd-party-mods-and-curse-voice http://boards.na.leagueoflegends.com/en/c/riot-official/EbE4QMZW-an-update-on-3rd-party-mods-and-curse-voice?show=flat&comment=000f http://boards.na.leagueoflegends.com/en/c/help-support/VgxepOm3-custom-announcer-voicepacks-and-third-party-programs?comment=0000

modifying the client in any way can result in perma ban

All modifications to the game are "use at your own risk". There are specific ones they have banned and people have posted a lot of information regarding that.