There are plenty of other guides out there, so this is more a guide for efficiency. Many thanks to WallRunnrNinja for getting me started on the roadmap.
First Playthrough
For my first playthrough, I did absolutely everything. Found every spell I could, beat every boss, gained every missable, gestured every gesture... did everything. This included using a single Bonfire Ascetic on the Primal Bonfire guardians in order to take their NG+ soul to Straid for the exclusive spell.
With everything I could possibly do, already done, I wanted to speed through the subsequent playthrus as quickly as possible.
First, make sure you have a bunch of Bonfire Ascetics available. There are plenty to find and a number of merchants have them for sale. Make sure you buy a bunch before you start the next play.
Second is "nice to have", but certainly helps with the soul gain. Have a full set of gear that boosts your souls earned when you kill enemies. I used:
Head - Warlocks Mask (Fragrant Branch in Shaded Ruins) - 2.5% Soul Bonus, also boosts Intelligence slightly. Can also use Nahr Alma Hood if you don't have a spare Fragrant Branch (see "Chest")
Chest - Nahr Alma Robes - 10% Soul Bonus (Kill Titchy Gren behind the Executioner's Chariot then buy from Merchant Hag).
Gloves - Jester's Gloves - 10% Soul Bonus (Buy from Magerold in Iron Keep). Can also use Tseldora's Manchettes (see" Legs") but Jester's offers much better protection.
Legs - Tseldora's Trousers - 5% Soul Bonus (Kill Maughlin in Majula).
Ring - Covetous Silver Serpent Ring +1 - 20% Soul Bonus (Spend 10,000 at Merchant Hag then Talk to her). The +2 ring is better (30%), but would need to Ascetic the Mytha boss fight. I chose to save it for next playthrough.
Second Playthrough
The first objective is to reach a 2 Million Soul Memory to open the door at the Shrine of Winter instead of lighting the 4 Primal bonfires. The second objective is to have enough Hard souls in your inventory to reach 3 Million Soul Memory in the Third Playthrough.
Before you go too far, check your Soul Memory. Its the number in top-right corner of your Stats screen. You want to have 2 Million more than this number.
My roadmap was as follows:
Things Betwixt (Just run straight through)
Majula
Forest of Fallen Giants, Last Giant Boss (Soldier Key, need it later)
From here I wanted to pick up the Covetous Silver Serpent Ring +2 and fight Old Iron King, but repeatedly fighting Rotten works just fine too.
Heide’s Tower of Flame, Dragonrider Boss
Exhaust dialogue with Licia of Lindeldt
Majula, pay Licia to move path
Huntsman’s Copse, Skeleton Lords Boss
Harvest Valley, Covetous Demon Boss
Earthen Peak, Mytha Boss (Equip Covetous Silver +2 Ring)
Iron Keep, Old Iron King (You can skip Smelter Demon)
From here I used those Bonfire Ascetics to repeatedly fight the Old Iron King. With all the Soul Bonus gear equipped, I would gain around 250,000 souls per kill, as well as two Hard souls - Old Iron King Soul (25,000) and Old King Soul (75,000). The Old King Soul is one of the most valuable of the Hard souls in the game. The Old Dead One's Soul dropped by Rotten gives 60,000, so the difference is negligible if you prefer this fight, but as a Sorcerer, Old Iron King was easier for me.
After roughly 5-6 fights, I had the 2 Million Soul Memory needed to open the door at the Shrine of Winter.
Majula, use Fragrant Branch of Yore on Rosabeth
Shaded Woods
Shrine of Winter
Drangelic Castle, Dragonrider Bosses and Looking Glass Boss
Shrine of Amana, Demon of Song Boss
Undead Crypt, Velstadt Boss
Obtain the King’s Ring
Drangelic Castle (King's Gate)
Throne of Want, Defender and Watcher Boss (to avoid fighting them just before Nashandra)
Shaded Woods
Aldia’s Keep (ignore torches, run past the dragon skeleton and down the hallway), Guardian Dragon Boss
Dragon Aerie (ignore everything, go across the zipline and run across the bridge)
Dragon Shrine (ignore most the enemies)
Speak with the Ancient Dragon to obtain the Ashen Mist Heart
Forest of Fallen Giants (Use Soldier Key)
Memory of Jeigh (Below Cardinal Bonfire), Giant Lord Boss (Giant's Kinship)
Throne of Want, Nashandra Boss, NG++ Unlocked
BEFORE you transition to your third playthrough, you want to collect as many Hard souls as possible so you can reach the 3 Million Soul Memory marker unlocking the Shrine of Winter. To do this, you can gain a Soul of a Great Hero (20,000) every 60 seconds.
Travel to Dragon Aerie bonfire
Run straight ahead and over the bridge (use Fragrant Branch)
Collect the Bonfire Ascetic, return to bonfire (Aged Feather)
Climb the ladder, use the Zipline and drop off after the bridge onto the island with the Soul of a Great Hero.
Return to bonfire (Aged Feather), use the Bonfire Ascetic
Once you run that a few times it will only take 1 minute to gain a new Soul of a Great Hero, for as many times as you need it. Only one Fragrant Branch is needed, this persists through the Ascetic. The amount of times you need to do this run depends on how many other souls you have. For me, with all the other souls I had, I only needed 50 Soul of a Great Hero, which took just under an hour. To give you an idea:
99x Soul of a Great Hero = 1,980,000 Souls
6x Old Iron King Soul + Old King Soul = 600,000 Souls
Once you have 3 Million worth of Hard souls, you're ready to speed through your final playthrough, which should take around 30 minutes. Check your Soul Memory before you do, just to be sure.
Final Playthrough
Things Betwixt (Just run straight through)
Majula
Forest of Fallen Giants, buy Fragrant Branch from Merchant Hag (12,000). Use Hard Souls if needs be.
Majula, use ALL your Hard Souls. Check your Soul Memory to make sure it exceeds the required 3 Million.
If for any reason you have come up short, don't fret. Pick the boss of your choosing and don your Soul Bonus gear and get to fighting. Rotten is nice and quick to get to.
Use Fragrant Branch of Yore on Rosabeth
Shaded Woods
Shrine of Winter
Drangleic Castle, talk to Chancellor Wellager to buy the remaining 2 Miracles and Pyromancy.
Achievement(s) unlocked!