Return to the foresty area of the Giant Chasm to see that it has mostly recovered, except for the sandy "crater" in the middle. Head to the eastern side of the forest, then north to reach Kyurem's cavern.
All the way back in the room where you fought against Black Kyurem or White Kyurem and Ghetsis, you'll find Kyurem, just sitting there, waiting for you. Make sure you save your game before fighting it, otherwise you'll have to wait until after you've beaten the Elite Four and Champion again for it to reappear, and they're pretty hard!
Unlike Zekrom and Reshiram, Kyurem has the standard catch rate of a legendary Pokemon, meaning it's likely to be a pain to catch. Dusk Balls are most effective against it for the first several turns, thanks to being in a cave, while Timer Balls are more effective after ten turns have elapsed.
You'll need to weaken it as much as possible in order to have a fair chance of catching it, and past that, it all boils down to luck, so you may catch it in your first Dusk Ball or it may take thirty (or more!). I recommend bringing about 10 Dusk Balls and 30 Timer Balls just in case.
Any Steel-type Pokemon (that isn't part Ground-type) should have no trouble walling Kyurem's attacks. It also has only two attacks that actually do damage, so you're at less risk than you were fighting Zekrom or Reshiram. Speaking of which, Reshiram is a particularly good Pokemon to use against Kyurem, particularly if you lead with Imprison to prevent Kyurem from using Dragon Pulse. Just be careful, as Reshiram's Dragon Pulse can possibly do enough to knock it out (although it will usually come close, which is good).
After catching it (or knocking it out), Cheren and Bianca will appear, say some stuff, and then leave. Make sure that you pick up the DNA Splicers that appeared where Kyurem once stood, which is the item you use to fuse Kyurem with either Zekrom or Reshiram!
In order to get yourself either Black Kyurem or White Kyurem, you're going to need to have both Kyurem and either Zekrom for Black Kyurem or Reshiram for White Kyurem, as well as the DNA Splicers item found after catching/defeating Kyurem.
You need to have both Kyurem and Zekrom/Reshiram in your party in order to fuse them. Use the DNA Splicers from your items (it's a Key Item) and you'll first need to select Kyurem, then you will need to select Zekrom/Reshiram for the second Pokemon. It will then fuse them together to create either Black Kyurem or White Kyurem, removing the other two Pokemon from your party (so you'll have an empty spot).
The fused form of Kyurem takes on the level, nature, and other details (like Effort Values and Individual Values) of Kyurem. Its moves may vary depending on which TM moves you've given it in either its Normal Kyurem form or Black/White Kyurem form, so keep that in mind when you fuse it or separate it.
You cannot use the DNA Splicers to fuse Kyurem if you already have another Kyurem form present, meaning you can't have both Black Kyurem and White Kyurem together. You also can't trade White Kyurem or Black Kyurem and need to defuse it first.
Additionally, after you've caught Kyurem, the next time you use Amanita's PC, you'll receive some special Kyurem wallpapers. Nifty!