Instructions
Stock up on Pokeballs. You need to have a good supply of the best Pokeballs you can afford with you at all times. If you run across a Shiny Pokemon, it might be your only chance to capture one, and if it escapes uncaught you have to start over. If you have a Masterball you might want to save it for just this purpose.
Ride you bike. While riding your movement speed increases and allows you to cover more ground, giving you an increased chance of encountering wild Pokemon.
Place a Pokemon with the Illuminate ability at the head of your party. As long as that Pokemon occupies that position, you will meet wild Pokemon more frequently.
Frequent areas populated by low-level Pokemon. When you encounter a Shiny Pokemon, this increases your chances of catching it without engaging in a risky battle that might render the Shiny creature unconscious and unable to be captured.
Be patient. You might need to encounter thousands of Pokemon before meeting a Shiny creature. The odds are constant for every single encounter.