Umbreon is one of the toughest walls to bring down thanks to its stellar defensive stats and access to Charm which allows it to handle most physical threats, including the omnipresent Curse sweepers. It's also a solid Pursuit user despite being very weak offensively since it doesn't fear much from Starmie and Dynamic Punch variants of Gengar unlike its two main competitors, Tyranitar and Houndoom. Umbreon can also act as a trap passer with the combination of both Mean Look and Baton pass which is potentially very rewarding if the opponent's phazer is out.
However, with such offensive stats and movepool, Umbreon may be the most passive Pokemon in the tier and rely on Toxic to do any sort of damage, which means it can't pose a threat to anything immune to poison except Gengar or able to heal it off with Rest. Umbreon itself despises statues heavily and has to use Rest in order to escape from poison and paralysis, and whereas Umbreon is extremely sturdy, it can be overwhelmed much more easily when it is asleep.
Set 1: Charmer
Move 1: Charm
Move 2: Pursuit
Move 3: Toxic
Move 4: Rest / Moonlight
Charm is one of the main reason why Umbreon is so hard to break through. The purpose of this move is to stop Curse sweepers like Snorlax in their track, essentially because Charm has twice the PP of Curse, but it has plenty of other uses : you can soften Explosion and powerful hits from slower Pokemon like Exeggutor and even Machamp. Beware of the high critical rate of Cross Chop in the case of the second one though. The threat of Charm means that even faster physical attackers and exploders like Heracross and Cloyster have to be careful when they're trying to switch on Umbreon, because they won't be able to pressure it if their Attack stat is reduced. Pursuit allows you to inflict significant damages to the likes of Exeggutor, Gengar, Starmie and Misdreavus if they try to switch-in, something they should do most of the time because the best they can do against Umbreon is either using Explosion, statues or Destiny Bond, respectively. Toxic is the best way Umbreon has to weaken the opponent and to force switchs and allows Umbreon to effectively check most non-Rest variants of Snorlax and Tyranitar. In conjuction with Pursuit and Spikes support, it can wear down some Pokemon pretty easily. Finally, Rest is Umbreon's preferred recovery option since it allows it to get rid of the potential poison or paralysis which either prevent it to stall or to make a good use of Charm. Moonlight is a fine alternative if you think you can avoid statues, however Umbreon won't be able to check threats like Body Slam Snorlax as effectively anymore, and the low PP is really awkward for an entirely stall-oriented Pokemon.
Outisde of its defensive utility, the main reason to use Umbreon is its ability to reliably weaken any Psychic and Ghost type with Pursuit, something greatly appreciated by Pokemon such as Machamp or mono-attacking Snorlax. Umbreon is also a good pick on Spikes-based teams since it's the only viable Pursuit user which fairs well against the arguably best spinner, Starmie, even though this one doesn't have much trouble recovering afterwards since Umbreon doesn't hit hard enough to get rid of it, and you have to keep an eye on your PPs while trying to weaken Starmie. Nonetheless, the star won't escape from Umbreon without being intoxicated or significantly damaged most of the time. Speaking of spinners, both Cloyster and Forretress are annoyed by Gengar and Misdreavus, so once again Pursuit is a great help.
Set 2: Mean Look Pass
Move 1: Mean Look
Move 2: Baton Pass
Move 3: Charm / Confuse Ray / Toxic
Move 4: Rest / Moonlight
With Mean Look and Baton Pass, Umbreon works as the bulkiest trap passer available. Charm still remains very useful against a ton of physical attacker, but here you have Mean Look which prevents the opposing Pokemon to escape while its Attack stat keeps decreasing. Once you have used Mean Look and potentially Charm several times, use Baton Pass in order to bring an appropriate counter to the trapped Pokemon or something which can safely set up on it and attempt a sweep. Confuse Ray is the best option Umbreon has against the omnipresent phazers which makes this set nearly useless and can be extremely frustrating for your opponent, especially when used in tandem with Mean Look since the opposing Pokemon will stay confused if it doesn't switch. However it's an unreliable strategy since it's based on luck. Toxic can still find some use to inflict residual damages, it works especially well against phazers like Tyranitar, Raikou and Suicune, because they will be either crippled or forced to use Rest, thus they won't be able to stand in the way of Umbreon consistently. Rest remains the healing move of choice because Umbreon still despises statues, although this variant isn't as stall-based as the former which means Umbreon can opt for a more flexible option in the form of Moonlight more easily, but besides the inability to heal off statues, the low PP is still an issue if Umbreon has to come multiple times throughout the game. However, you can forget the healing move in favor of another move listed in the third slot if you really the need to carry two moves to disable your opponent.
As already said, Umbreon can't transfer Mean Look if there is at least one Pokemon able to phaze it in the opposing side, the most common ones being Skarmory, Raikou, Suicune, Steelix and Tyranitar. Thus, Pokemon that can lure most of them like Lovely Kiss or All-out Attacker Snorlax makes good partners, keep in mind that the first three need to use Rest in order to heal themselves and they will rarely carry both Sleep Talk and Roar on the same moveset. As a consequence forcing them to do so by weakening them is also a good way to prevent them from disturbing Umbreon. Finally, sturdy sweepers like Curse Snorlax or Vaporeon makes good teamates since Umbreon can give them many opportunities to set-up.
Umbreon learns Growth and can transfer it via Baton Pass, and while it's still very weak even boosted unlike it's brothers Vaporeon, Jolteon and Espeon, it can live much longer thanks to his bulk, typing and access to Moonlight and can still dent Skarmory and Steelix with a +1 Hidden Power Fire. Umbreon also has access to both Curse and Substitute, but don't have much tool to use in tandem with them in order to prevent phazers from doing their job. Zap Cannon is an option in both set to spread paralysis and to hit some annoyances like Skarmory, but the Toxic is usually the best option for such a stall-based Pokemon. Sleep Talk is here if you need to keep using Umbreon while it is asleep, but it's really hard to fit. Sand Attack is an alternative to Confuse Ray in the second set to mess with the opposing Pokemon by nullifying it's accuracy, but it's often even more unreliable than just confusing it in one turn.
[Checks and Counters]
When you don't know which Umbreon's variant you're facing, the safest switch-in at your disposal are probably phazers immune to Toxic like Skarmory or Steelix, those which can cure of poison via Rest like Raikou, Suicune and Tyranitar are fine alternatives too. Heracross is one of Umbreon's biggest fear : all it needs is to avoid Charm on the switch-in in order to put a big dent with Megahorn, and it will carry Rest most of the time so Toxic won't disable it too much. Faster exploders like Cloyster and to a lesser extent Gengar are in the same vein but are obviously less reliable answers since they have to sacrifice themselves to weaken Umbreon. Most set-up sweepers can't take advantage of Umbreon's passivity because of Charm, but they are some exceptions like Growth Vaporeon which will boost its Special attack in front of Umbreon fearlessly. Other examples are the uncommon Substitute users like Tentacruel and Kangaskhan since their sub needs at least three Pursuit to break. Belly Drum Snorlax is also a threat if Umbreon try to come in the turn he boosts himself, but he shouldn't try to directly set up in front of the cat as it will get charmed twice before being able to hit it. A common way to bypass Umbreon is to use statues as a paralysed one is much more easy to break through for powerful attackers like Marowak and especially Machamp whereas an intoxicated one won't be able to stall against anything. Thus it will have no choice but to use Rest and try to stay healthy during two turn while being asleep. Outside of phazers, opposing Pokemon with Baton Pass or Perish Song are big annoyance for the Mean Look variant. Once again, Pokemon immune to Toxic or with Rest doesn't fear anything from Umbreon, although they can't necessarily do much in return.