To acquire the Art of Abysswalking, you need to defeat the great wolf Sif, who is located in the Darkroot Garden. You can buy the Astoria seal from the Undead Parish blacksmith to unlock the large sealed door guarding the area. Alternatively, you can go around the back, and follow the shoreline that harbors the Hydra (Near the giants, and the Havel tower). Sif is located just past the Forest Hunter covenant.
You can enter the Darkroot Garden area, through the large doorway adjacent to the Undead Parish blacksmith. The area is guarded by a Titanium Demon.
Strategy on defeating Sif:
It is a good idea to bring a shield to this fight; With enough endurance, all of Sif's sword attacks can be blocked. The generally accepted strategy for defeating Sif, is to stay under his stomach, and strike when you are safe from his sword attacks. Like most bosses, you just need learn the patterns, and the rest will come with practice. Good luck.
Where to use the abyss ring:
The abyss ring is used to enter the domain of the the Four Kings, near the end of the New Londo Ruins area. In the area filled with Dark-Wraiths (big black-armored knights), you will find a white gate that takes you to a spiral staircase, leading down into a pitch black oblivion. If you don't have the abyss ring equipped while descending, you will die. When you have the ring equipped, you will fall into the abyss, but you will survive. The fall, at least.