Kyogre is a bastard and incredibly hard to take down. That massive Special Attack stat gives its STAB and rain-boosted Water-type moves a real kick.
Substituting Kyogre. Thunder is nice for other Kyogre and paralysis, Ice Beam for Groudon (in the sun), Rayquaza, Grass-types, and primarily, Latias and Latios. Calm Mind boosts Kyogre's already amazing Special Defense and makes it nearly unbeatable by Electrics. Knowing that most physical sweepers are countered by Water Pokémon, you are going to have a tad of trouble killing this thing with a few Calm Minds under its belt. 101 HP Substitutes block Blissey and Deoxys-D while you Calm Mind up. It takes 3 Calm Minds to 2HKO Blissey, but watch out for Psych Up and Calm Mind on Blissey (only if it has Thunder).
Kyogre is one of those Pokémon that has so much power that it can make use of stuff like the berries, unlike most Pokémon. After a Salac Berry and a few Calm Minds, Kyogre can be quite annoying... It's too slow to use Petaya effectively (the slowest Pokémon in Ubers, really, except for Metagross, Blissey, and Deoxys-D).
The other main set. Without Substitute, you can have more diversity in attacking.
Rest keeps you safe from Toxic, paralysis, and other random forms of damage as you Calm Mind up for the sweep. With these EVs, you survive a Swords Dance Groudon Earthquake, and OHKO it in the rain without a Calm Mind. Just about the only thing that can stop this is another Kyogre with Calm Mind and Thunder, or Calm Mind Latias / Latios. Most other things fail to KO it after a Calm Mind. It easily beats any enemy Kyogre without Thunder or Rest, as well.
Water Spout is the overlooked move for Kyogre, and is often seen as amateur. Rest allows Kyogre to bring Water Spout back to full power, and Sleep Talk lets you do some damage while Sleeping, so you don't waste all of Water Spout's potential on those Resting turns. This Kyogre almost requires paralysis support, as Kyogre is slow, and taking hits is one thing you don't want to do. With paralysis, however, Kyogre can get in a hit before taking one, using a 150 power STAB rain boosted Water move. Water Spout does ~52% to Latios, letting Ice Beam finish it off. It can OHKO Groudon in the sun, and does 44% or so to most Uber Blissey, meaning that with a few layers of Spikes down or if they lack Special Defense, you 2HKO.
This set is surprisingly effective, unlike its counterpart, the special Groudon. The three most common counters to Kyogre are Blissey, Latias, and Latios. Hidden Power Ghost 2HKOs Latias and Latios, regardless of Calm Minds, and Brick Break 1-2HKOs Blissey. It's best to open with Surf or Ice Beam, usually, unless you know your opponent's team enough to predict what they'll switch to, as this set does benefit a lot from that turn or two of surprise. However, even once they know what kind of Kyogre you have, many teams have a dickens of a time countering it anyway. Unlike Groudon, Kyogre resists Ice Beam, which is far more common than Thunder, and, thanks to its superior Special Defense, takes less from Thunder than Groudon does from Ice Beam. Kyogre's typing also allows it to come in on things like Meteor Mash.
Surf and Ice Beam are to stop certain Pokémon from walling you. Surf hits Groudon in the rain with an OHKO, while Ice Beam is stronger in the sun. Ice Beam will OHKO Rayquaza, Surf hits Skarmory and Metagross. Rock Slide can be used to hit Rayquaza if you're using Surf, as well as Ho-Oh, but Sleep Talk makes Kyogre a perfect Exeggutor counter, and eliminates another weapon from Blissey's arsenal (Sing). Earthquake can be used, as well, but it won't be hitting much, if anything, harder than your other moves, aside from the rare Lanturn.
Hidden Power Dark and AncientPower are both for killing Shedinja. Safeguard is decent, but not really that great of an option for a sweeper like Kyogre. With Endure, it can take a Swords Danced Groudon Earthquake and then use the extra Speed from a Salac Berry to sweep the team with a Calm Minded, Rain Danced Surf.
Substitute versions need max HP. Otherwise, concentrate on Special Attack. If you really feel like it, with max HP and around 280 Defense, Kyogre can take a Swords Danced Groudon Earthquake. There are two main ways to do the Speed. If Kyogre is your lead, you'll most likely want to make it as slow as possible, perhaps even giving it a -Speed nature and 0 IVs. This is so that when you are up against starter Groudon, your weather can go after theirs, thus claiming priority, forcing a switch. Non-boosted Groudon will not OHKO Kyogre, while Kyogre OHKOes Groudon in the rain with any Water move. As Kyogre is already tied for slowest Uber (and all Pokémon used in Ubers slower than Kyogre are slower, regardless of EVs, for example, Blissey), Speed won't really change much. However, since a lot of people think this way, it's very easy to beat out enemy Kyogre, Groudon, a lot of Lugia, Adamant Rayquaza, and even some Latias and Latios.
Checks and Counters
Blissey with max Special Defense (unless it has Substitute or Rest), or Calm Mind / Psych Up + Thunder, Hidden Power Grass Ludicolo, Quagsire (especially for sets with only Thunder and Surf), Shedinja (beware of Spikes), Latias, Celebi (if it doesn't have Ice Beam), Lanturn with Toxic or Mimic.
Controlling the weather
In a match-up of Kyogre vs. Groudon, if both Pokémon are the first each battler sends out, the slower Pokémon will always control the weather. This is because the faster Pokémon's weather activates first, but is promptly overwritten by the slower Pokémon's.
If Kyogre switches into Groudon, or vice versa, the weather will change.
If Kyogre and Groudon switch into each other at the same time, then the person who makes the decision last switches in their Pokémon last, thus dominating the skies. In matches where a person has a limited amount of time to move, this can cause the players to try and play chicken, waiting until the clock is as close to running out as they dare before punching in the move, and praying the connection doesn't lag.
Rayquaza / Golduck
If Rayquaza or Cloud Nine Golduck is out, it will report that the weather is still going strong. However, there will be no effect. SolarBeam takes 2 turns, Thunder has 70% accuracy, and no moves are boosted / weakened. If Kyogre or Groudon switch into Rayquaza, Air Lock will still be in play. Once Rayquaza leaves the field, the weather returns to normal.