*This solution contains spoilers.
Finding all the pyromancy spells can be difficult because you are required to not only acquire the Pyromancy Catalyst if you are not a Pyromancer, but also dole out all the souls required to upgrade it to +10. This is needed because a certain Pyromancer NPC, Quelana is Izalith, will not show up until you do that.
First off, I'll list all the NPC's required for this and how to interact with them.
-Laurentius of the Great Swamp: Laurentius sells many of the "basic" pyromancy spells. The thing is, this NPC is completely missable. You could even kill him by accident if you are careless. You'll find him in a room in the Depths just after you kill the first Butcher. He is trapped in a room full of barrels. He is actually stuck in one of these barrels in a corner. Do not hit the barrels with your weapons to free him. This risks you hitting him and turning him hostile forever. Instead, roll into the barrels to break him loose. He'll then head to Firelink Shrine. Go on to defeat Quelaag in Blighttown Swamp. Immediately after this, head to Firelink Shrine and talk to Laurentius. Level up your Catalyst to +10 and buy any spells you do not own. After a certain amount of time, or if you talk to him after encountering Quelana, Laurentius will go to Blighttown and turn Hollow, barring you from interacting with him anymore. Aside from you stabbing/slicing/piercing/spelling him to death.
Spells bought from Laurentius:
-Eingyi: Eingyi is located in Quelaag's Domain behind a secret wall. He is an egg carrier, but he is not hostile. Answer "Yes" to the question he asks and you will be let into the chamber that has a covenant leader as well, Daughter of Chaos. Eingyi will not sell you any spells unless you have been infected with the egg status. To get this, attack an Egg-Carrier enemy in Qualaag's Domain to make it grab you. When this happens, you'll be infected and eventually, eggs will grow on you. Eingyi will now sell you his spells. As well as egg vermifuge, the item that gets rid of the disgusting beast now growing on you. He also ascends your Pyromancy Catalyst if somehow you saved Laurentius, but you can no longer use him.
Spells bought from Eingyi:
-Daughter of Chaos: She is located in the same room as Eingyi. You need to join her covenant if you want to get all the spells. Upon joining, you receive Great Chaos Fireball. To get her other spell, you need level up her covenant by offering humanity. Upon reaching level 2, you will receive the spell, Chaos Storm.
-Quelana of Izalith: Quelana will show herself in Blighttown Swamp once you have done several things. You must have saved Laurentius, defeated Quelaag, rang the second bell, and you must have leveled your Pyromancy Hand to +10. Once you have done this, Quelana can be find in Blighttown Swamp, sitting next to one of the large stone pillars near the entrance to Quelaag's Domain. She will sell you several spells and even level your Pyromancy Hand to +15, then ascend it to an even stronger version. Once you have placed the Lordvessel on the Firelink Altar, return to Quelana. She will give you a special mission to kill her mother, Bed of Chaos, the boss of Lost Izalith. Once you've completed this, she'll give you the incredible Fire Tempest spell.
Spells bought from Quelana of Izalith:
These spells must be found.
Dropped by Engorged Zombie in the Painted World of Ariamis.
Found on a corpse in Painted World of Ariamis.
Found on a corpse in Blighttown.
-Chaos Fire Whip
Found in a chest in Lost Izalith.
Good luck! Any and all comments/questions are welcome!