The two primary methods for shiny hunting in X/Y is by using the Pokeradar and horde encounters. However, Eevee can't be found in hordes so the Pokeradar would obviously be the prime choice. To give an in depth explanation to the mechanics of the Pokeradar would use up a lot of space and has been explained before. Instead I will leave you with a link from Smogon explaining how to use it to chain for shinies here.
Now, chaining with this method has its risks, and personally I find it very frustrating. In my opinion the best way to find shinies is through the Masuda Method. Through this method there is less risk of losing your shiny and as a bonus you can breed down any Nature, IVs, and egg-moves you may want. You will need a Pokemon (normally a Ditto) from a version of the game in a different country. This can be easily obtained through the GTS or can be arranged with people online. Breeding Pokemon from two different countries increases the chance of finding a shiny.
Through the Pokeradar your odds will increase from 1/4096 to 40/4096 (or around 1/200)
- You can use the Cute Charm trick to find a specific gender
- Higher odds of encountering a shiny
- Chance of breaking your chain/KOing the shiny
Through the Masuda Method, your odds will increase from 1/4096 to 6/4096 (or around 1/683)
- Any Nature, IVs, and/or egg-moves you want
- More reliable way to make sure you don't lose your shiny
- Lower odds of encountering a shiny
Do some research and experiment with each method to find the one that works best for you.
Hope I helped!