Do not kill Quelaag's sister. After defeating Chaos Witch Quelaag, enter the circular room beneath the Bell of Awakening, with cobwebs lining eggs on the sides. Pass through the illusory wall to the right to find an egg-infected man. Answer "Yes" to his question and he will move. You can also kill him and join without repercussions but this will cost you his services. Talking to Quelaag's Sister will give you an option to join the covenant.
Note: Giving the Lordvessel to Frampt blocks you from entering this covenant in your current game session. Accessing the Darkwraith covenant requires being able to speak with Kaathe. To access Kaathe, the Covenant of Artorias ring is required (dropped from the Great Grey Wolf Sif), as well as the Key to the Seal (obtained from Ingward), and defeating the Four Kings. After defeating the Four Kings, Kaathe will appear. Answer "Yes" to his question and he will teleport you to the Firelink Altar. Speak to him again in the Abyss after this event to join the Darkwraiths.
You must not have attacked Alvina. To open the door leading to the Alvina, purchase the Crest of Artorias from Andre of Astora in Undead Parish. It is also possible to reach the covenant without opening the door, but this will require significantly more time and defeating more enemies. Find Alvina at the Darkroot Garden, standing just before the bridge linking to the ruined structure. Answer "Yes" to her questions.
Do not kill Gravelord Nito and obtain at least one Eye of Death. You can find some Eyes of Death near to where you can access the Gravelord Servant covenant. They are also dropped rarely from Basilisks in the Depths and The Great Hollow. Then, find the area in The Catacombs with a row of coffins guarded by a Titanite Demon. One coffin near the end of the hall, before awakening the Titanite Demon, has an open lid. Enter it. After about thirty seconds, if you do not have active phantoms, a short cinematic will show you being teleported to an enclosed room. Pray to the "Sarcophagus Gravelord" there.
Talk to the Stone Dragon at Ash Lake. To reach Ash Lake, travel from Blighttown to the The Great Hollow. Move to the very bottom then follow the path along the beach until it ends. The entrance to The Great Hollow is behind an illusory wall on the left side of the Blighttown swamp.
Talk to Gwynevere after receiving the Lordvessel from her in Anor Londo.
You must have 25 Faith. This number is lowered by 5 for every successful co-op Boss kill as a white phantom. You can then kneel in front of the broken statue on the balcony behind the Hellkite Dragon.
Talk to either Petrus at the Firelink Shrine or Rhea after you have rescued her from the Tomb of the Giants. Note: If you chose to start as a Knight or Cleric, you will already be in the Way of White but this will not earn "Covenant: Way of White".
Exit the arena where the Chosen Undead felled Gwyn instead of lighting the bonfire.
Crossbreed Priscilla is an optional Boss in the Painted World of Ariamis. She initially asks you to leave the world without fighting. To reach the Painted World, collect the Peculiar Doll found in the Northern Undead Asylum, upon returning there later in the game. When searching the large guarded painting in Anor Londo, you will get pulled into a cloistered area. To return to the Asylum, climb to the top of the ruins in Firelink Shrine and reach the Giant Crow's nest.