On the current Topic: I own both an attacking Umbreon and a defending Umbreon, so I should already be good at this.
-Payback/Dark Pulse. Payback would usually be a bit stronger but Dark Pulse can flinch opponents. Payback's power could go up to 125 or 150 (Depending on when STAB occurs) but Dark Pulse could only get up to 120. Payback isn't something you should rely on all Pokémon, as there are Pokémon slower than Umbreon. Dark Pulse could get you another chance to attack if the opponent flinches. Choose Payback if you choose Curse (See below) but choose Dark Pulse if you want some extra Attacking chances.
-Psychic. You would want to take advantage on the ONLY good Psychic-Type move Umbreon could learn. It's got 90 power, which is strong already, and a slight chance of lowering Special Attack. You should take this move into consideration if you want Umbreon to get more and more powerful when using Psychic (Considering Special Defense is lowered). Also, with this move, you don't need to worry with most Fighting-Types no more (Especially since they usually have bad Defenses).
-Confuse Ray. With this, almost NOTHING can stop Umbreon. Mainly because once a Pokémon is confused, they have 50% chance of hurting themselves in their confusion (Which has a Base Power of 40, and is a Physical Attack). This makes Umbreon more Offensive AND Defensive at the same time, and confusion rather lasts quite a long time (Especially if the Pokémon is charging up after an attack or couldn't move from Paralysis). This kind of gets rid of all of Umbreon's weaknesses by itself. However, it obviously does not affect Normal-Types.
-Curse/Moonlight. Curse is good to increase Physical Attack and Defense in battle. This makes Payback (If you choose it) very strong. After it is used, Umbreon will be very hard to beat. Moonlight increases Umbreon's HP, depending on the time of day and what conditions are occurring. Moonlight works best if used in Sun in a battle, restoring Umbreon's HP by 2/3. It is still useful if no weather is happening in a battle, but only by 1/2. Choose Curse if you're going to choose Payback but choose Moonlight if you choose Dark Pulse.
For my Attacking Umbreon, the Move-Set was slightly different. It is very strong and resists everything. Synchronize helps Umbreon in offense a lot.