Flygon has a unique typing; its resistance to Rock and immunity to Ground and Electric makes it a solid check to Tyranitar, Aerodactyl, Hidden Power Grass Raikou, and others. Its ability also means it's the only Rock-type resist immune to Spikes outside of Claydol. The fact that it gets STAB on Earthquake is also nice; it has very good Attack and Special Attack stats as well. In fact, it has decent stats all around, and can act as a makeshift tank in a pinch. All in all, it is very good Pokemon that can and should be used in the OU metagame. If you use it well, it should bring you a lot of success.
Flygon's typing and ability allows it to check many Pokemon, and this set takes advantage of it. Earthquake is your STAB and is Flygon's most threatening attack. Rock Slide takes out Aerodactyl and Zapdos, both of which are kept in check by Flygon's typing. It's also Flygon's strongest attack against threats like Gengar. Fire Blast keeps Skarmory and Forretress from walling Flygon, while Hidden Power Bug allows it to take out Celebi and is Flygon's strongest attack versus Claydol and opposing Flygon. Toxic can be used over Hidden Power Bug for Swampert, which normally keeps Flygon in check. It also severely hinders common switch-ins such as Porygon2 and Claydol.
With 188 HP EVs, Flygon is never 2HKOed by standard Dragon Dance Tyranitar's Hidden Power Bug, making Flygon a complete counter to that variation of Tyranitar. 140 Speed EVs give Flygon 270 Speed, enough to outspeed Adamant Heracross and max Speed Modest Suicune. If you're wondering why to not alter Flygon's IVs to give it a 31 Speed IV Hidden Power Bug, it's because the 31 Speed variant drops Flygon's Attack and Defense, both of which are needed. A Naughty nature rather than Adamant allows Flygon to 2HKO defensive Skarmory and guarantees a 3HKO on specially defensive Skarmory.
An alternative spread of Naive with max Attack and Speed can be used for a more offensive Flygon. This spread always outspeeds Timid Hidden Power Fire Celebi and ties with Timid Jirachi. This spread loses the bulk of the first one for a chance to outspeed most Pokemon, including Timid Suicune.
The strategy here is simple: switch into either another Choice Band user or physical sweeper using Earthquake or Rock Slide, predict what they're going to switch to or if they'll stay in, and attack accordingly. As with all Choice Band users, this set requires good prediction, but has a good payoff. Hidden Power Bug is useful for Grass-types, Celebi in particular, while Fire Blast takes out Skarmory.
This Flygon will want to maximize its Speed in order to beat Timid Celebi, Salamence, Houndoom, and to tie with Jirachi. Like the previous set, a Naive nature rather than Jolly boosts Flygon's power against Skarmory. Naive is preferred over Hasty in order to check Tyranitar and Aerodactyl more effectively.
Flygon's typing makes Substitute a great idea for a Pokemon that can set it up so easily. Like the first set, you're mainly looking to rely on Earthquake and Rock Slide, but using more HP to set up a Substitute on weaker attacks. Zapdos's Hidden Power Grass will never break Flygon's Substitute and opposing Flygon will usually fail to break its Substitute as well.
Flygon usually checks itself well, so using Substitute in conjunction with Toxic can take down what might be their only answer to your Flygon. Toxic also wears down threats such as Swampert, which makes Flygon an excellent teammate to Pokemon like Tyranitar and Salamence. If Toxic seems too slow-paced for your team, Hidden Power Bug can be used to hit Celebi while Fire Blast prevents Skarmory and Forretress from setting up on Flygon's Substitutes. Another option is to partner Flygon with Magneton to deal with Skarmory and Forretress, rather than using a slot for Fire Blast.
A special set can use Dragon Claw, Fire Blast, and Giga Drain to take on defensive teams revolving around Swampert, Salamence, Skarmory, and other Pokemon that might give standard Flygon trouble. Hidden Power Ghost can wreck an incoming Gengar, while Hidden Power Flying OHKOes foolhardy Heracross trying to switch into Earthquake. Screech can force switches and threatens some Pokemon that Flygon couldn't normally take out, giving Flygon a chance to use its wide coverage or beat last Pokemon Snorlax or defensive Suicune.
Checks and Counters
Ice-type attacks are usually the surest way to deal with Flygon. Claydol and bulky Pokemon with Hidden Power Ice are very safe counters to it. Porygon2 can trace Levitate to evade Earthquake and is weak to none of Flygon's attacks. Swampert, Milotic, and Suicune can come in reliably on anything but a Choice Band-boosted Earthquake—which doesn't OHKO regardless—and use Ice Beam on Flygon. Heracross and Skarmory can deal with Flygon lacking Fire Blast, and can still survive a hit and force it out even if Flygon does carry it.
Celebi can take on Flygon that aren't using Hidden Power Bug, and Reflect Celebi can even hold its own versus Flygon that do. Cloyster and Gengar both take little damage from Flygon's attacks and can comfortably OHKO back. Be careful of Electric-types that hold Hidden Power Ice, as they can make short work out of Flygon that expect Hidden Power Grass. Intimidate from Salamence and Gyarados can weaken Rock Slide so that Flygon can barely touch them.