I have just completed the achievement Bond of a Pyromancer so I decided to write a quick guide on how I got all pyromancies.

This achievement can be completed entirely in NG. Certain parts must be done before the Bed of Chaos is defeated. It requires no boss souls, and there are also no stat requirements.

Note that the Black Flame pyromancy is NOT needed to get this achievement.

• Laurentius
  
• Quelana
  
• Eingyi
  
• Quelaag's sister (The Fair Lady)

• Iron Flesh (If Laurentius is told about Blighttown pyromancies)
  
• Flash Sweat (If Laurentius is told about Blighttown pyromancies)
  
• Fire Tempest (If the Bed of Chaos is killed before Quelana asks for a favor)

His Location: Laurientius is located in the depths in a room near the butcher. Approaching from Undead Burg, enter the depths into the butcherer's room. Drop down beneath the arches as if you were going further into the depths. To the left there will be a room, and Laurentius will call for help from a barrel. Roll into the barrels to break them and free him. Do not attack the barrels, as this may hit and aggro him. (If you do aggro him, request absolution at Oswald at the Bell Gargoyle tower.) He will relocate to Firelink Shrine.

What we want from him:
Pyromancy Flame
Pyromancies (Two unique)
+10 Pyromancy Flame

If you did not choose the pyromancer class, he will gift you a pyromancy flame. Then, you can buy the pyromancies he has to offer. Flash Sweat and Iron Flesh are unique to him and can only be bought from him.
If you already have a pyromancy from Quelana or Quelaag's sister, he will ask you to share their secrets with him. Answer NO. Answering yes will cause him to relocate to Blighttown and go hollow. If this happens before his unique pyromancies are bought, the achievement can not be done in this game cycle, and you must go into NG+ to repeat.

In order to get Quelana to spawn, we need to upgrade our pyromancy flame to +10. This does not require materials, just lots and lots of souls. If he has died or moved to Blighttown, this can still be done by Eingyi at the Chaos Servant Covenant.

Her location: After a +10 pyromancy flame is gained, she will spawn in Blighttown, sitting against the pillar that is directly between the sewer bonfire and Quelaag's domain. If the Bed of Chaos has been already defeated, she will not spawn.

She sells many unique pyromancies. Buy all of her pyromancy stock before the Bed of Chaos is defeated. She will then ask you a favor, that is to kill the Bed of Chaos. Once this is done, she will gift you Fire Tempest. If you have defeated it before she asked for the favor, killing her will drop Fire Tempest.

Location: Beneath the bell in Quelaag's domain, there is an illusionary wall. Give it a smack, and you will see a hallway leading to Eingyi. He will ask if you came to be a servant. Answer YES. If you answered no, just ask him again later.

Join the Chaos Covenant and you will get Great Chaos Fireball. Offer 30 soft humanity to receive Chaos Storm.

To get Eingyi's pyromanices, you must become infected with "egg-head." Go down to demon ruins and walk up to one of the crawling egg-carriers. Let them grab you a few times, then walk away and stand still. If your character scratches their head occasionally, you will get egg-head in a few minutes. Once you do, talk to Eingyi again and he will sell two unique pyromancies. (If you talk to him while egg-headed but don't buy anything, then come back later non-egged, he will still sell them)

That is all you must buy. Now find the few that are left laying in various locations:

Power Within - Blighttown (Underneath the Parasitic Wall Hugger)
Chaos Fire Whip - Chest in Lost Izalith
Acid Surge - In the graveyard area to the right side as you enter near the edge of the cliff in the Painted World of Ariamis.
Fire Surge - Drop from the passive Bloated Dredge located after passing two illusory walls under the well, in Painted World of Ariamis.

And that's it! As stated before, the Black Flame does NOT need to be obtained to get the achievement.

The order of the steps is not rigid. For example, I did the Chaos Covenant Pyromancies first, then Laurentius's, then Quelana's last. The order in this guide just guarantees that nothing will be missed.

Hope this helps! I enjoyed writing this much more than the homework I was supposed to be doing.