Instructions
Head to Hearthome City, the fourth city of Pokemon Platinum which can be found directly in the center of the Sinnoh region Town Map. Visit Bebe house next to the Pokemon Center. Bebe is the one who designed the Pokemon storage system in Sinnoh that allows you to store Pokemon as data on the computer system. If you have six Pokemon in your party deposit one into Bebe's Computer and then speak to Bebe. She will offer you a new Pokemon, the level 20 Eevee.
Place Eevee at the top of your team so that it is the first to be released. This will make the Pokemon more affectionate towards your character. The aim is to make Eevee as happy as you can so take it for a massage, treat it to Rare Candy and try to keep it from fainting in battle. Do whatever you can to raise its happiness. It can take some time, but your persistence will pay off eventually.
Train your Eevee by fighting weaker Pokemon in the wild or by challenging other trainers. Alternatively, have your Eevee enter into a battle against a stronger Pokemon and immediately switch it with another. By having the Pokemon in battle even for a short time will give it a great deal of experience points as the points are shared with every Pokemon that appears in battle. The more you battle the more its happiness will increase, but if you allow it to faint your Eevee will be less affectionate towards you.
Wait until night time. Pokemon Platinum version works off the Nintendo DS internal clock and different things can happen at night than during the day. In the case of Eevee's evolutions you can get a different Pokemon depending on what time of the day it is: in the daytime it will become Espeon, and at night it will become Umbreon. Make sure to level up at night time and you should successfully evolve your Eevee into an Umbreon.