Before you encounter the legendary Pokemon save right in front of it (if possible) and then encounter the Legendary. If it isn't shiny, restart your game by pressing : L+R+Start.
The odds are a really low 1/4096 Chance without the Shiny Charm.
You can hack your game with Powersaves and give yourself some shiny legendaries.
It could damage your Save File though, so be careful
GTS and maybe Wondertrade could reap you some Shiny Legendaries, although quite a few of the legendaries are likely to be hacked. Other places are the DB Chatroom, Reddit, GameFAQs and Pokemon Showdwon WIFi room. The Showdown WiFi room is a great place to start as illegally hacked Pokemon are banned from being traded Poketrade.us is also an option.
Good Luck with the shinies :D