Breeding is the best way, and Matsuda method is the most well-known of them. You take a female Eevee and put it in the Day Care with another male Pokemon of the Field Egg Group (such as Skitty or Wailord) making sure that the two are from different countries of origin. This increases the chance of a Shiny hatching from the Egg to six times the normal rate of 1/4096. If you have completed the National Dex, then you can obtain the Shiny Charm from Sycamore. This item triples the chances of getting a Shiny. Combined, they give you a 1/512 chance of a Shiny Eevee.
