To catch Kyurem, you first need to finish the main game. Then you need to reach Lacunosa Town, which you can reach by going east from Opelucid City.
Once in Lacunosa Town, put a pokémon in your party that knows Surf, and a pokémon that knows Strength.
Take the east exit of Lacunosa to reach Route 13. Go south and then west to find a set of stairs leading down. Go east from here, under a bridge, then Surf north to find some stairs. Go up the stairs. To the left there is a rock that you can push with Strength, so do that. Then go north. Follow the path to go into the cave entrance.
If you want to collect some items, go left from the cave entrance. Otherwise, go right from the cave entrance and use Strength to push the boulder. Then go left and down the stairs to reach the exit.
To make Kyurem appear, go down the stairs, then northwest, and drop down the ledge. Then go north and along the north wall, but skip the first ledge that you reach, and continue east to find another ledge. Drop down it. You will see some stairs, but you can't reach them right now, so just ignore them.
Go southeast and drop down another ledge. Go south and then west. You will pass by TM03 Psyshock. Don't drop down the ledge that is southwest of the Psyshock or you will have to start over from the beginning of the area. Instead, head northwest and drop down a ledge.
From there, go right (through a bunch of tall grass) and down. You will find a small pond. Walk up to it and you will hear a pokémon sound and it will start to snow. After that, the whole area is covered in snow. Go northeast to find some stairs.
Kyurem is to the north. Be sure to save your game before trying to catch it, because if you KO it and then save the game, you won't be able to encounter it again.
It's at level 75 when you encounter it. Its attacks are Glaciate, Dragon Pulse, and Endeavor, and it also knows Imprison.
The best pokéballs to use on it are Dusk Balls. If 30 turns pass, you can use Timer Balls for a slightly higher probability of catching it.
After you catch or KO Kyurem, the snow in the Crater Forest area will melt and you can access any items that you didn't pick up before.