Instructions
Go to the Cave of Origin, which is north of Sootopolis City.
Walk around the cave until you find Kyogre. He will be in the lake somewhere in the middle of the cave.
Do not forget to save your game before you try to face Kyogre. You may have to reload multiple times.
Press the "Action" button to begin the battle with Kyogre. Watch the orb asurrounding him before the battle. While he is normally blue, if he is shiny, he will have a purple hue and will flash briefly before you send your Pokemon out.
Reset the game if he isn't shiny by pressing "Select," "Start," "A" and "B" simultaneously.
Continue to reload and reset the game until you see a shiny Kyogre. Remember, your chances are very low, so it can take hours or even days to get one, but once you do you've got a very rare Pokemon.