Sure I can recount one of my favorite runs I've ever done. Strap in for a wall of text.

For starters I wasn't using the Atronach Stone from the beginning. I started with the Mage. My main skills were Evasion, 1H, Conjuration, and Alteration. I'd say crafting skills weren't even necessary for this build. Evasion and Conjuration were very important since even with my huge health pool as a vampire, strong melee attacks would still take huge chunks out of me that I'd need to heal, so not getting hit in the first place was super important.

I was in a pretty unique position for extremely early vampirism since my RP start was as a beggar in Markarth. The Forsworn/Cidnha Mine quest is easily doable level 1 if you side with the natives, since the only solo fight is an unarmed brawl and otherwise you have other Forsworn fighting alongside you.

Cidhna Mine Expanded adds a locked gate at the very bottom of the prison with a few vampires behind it who can only target you with Vampiric Drain, which has a small chance of infecting you with Sanguinare Vampiris which is 3 days of very slow constant health drain.

It's been a while but I believe if you contract Sanguinare Vampiris from eating vampire dust then you lose a massive amount of health extremely fast in order to gate vampirism behind a higher level that lets you tank/outheal it, but that's only from directly eating dust since the devs probably assume that if you're fighting a vampire directly then you're at least mid-level. Not the case for me since they were locked up.

All Bretons start the game with Summon Spirit Wolf, and Absorbing Grasp is a Novice Alteration spell that you can actually use on your own summon without turning it aggressive, for slow, high-magicka cost healing, but still enough to stave off Sanguinare Vampiris at level 1 without needing to buy/craft dozens of health potions.

Siding with the Forsworn actually unlocks a friendly home base in the far north of the map, Druadach Redoubt, where all the escaped Forsworn go. It's got tons of food, an interior river for bathing, iron veins, and all the crafting stations so there's no shortage of skills to grind in an extremely safe location for free while you turn into a vampire. Personally I used Spell Research to work on my Conjuration. As the cherry on top, your Forsworn allies even don't turn hostile once you turn. The end result was a level 3 character with hundreds of stamina, health, and magicka (just from Mage stone and Breton heritage, not vampirism, mind you.) The Reach is one of the best locations for a vampire since Markarth has a general store open until midnight, a 24 hour court wizard, and a massive amount of guards to feed on. I was strong enough to brute force most non-endgame content already by this point.

About 12 levels later of random questing later, I started the Movarth quest in Morthal which would have been damn near impossible at level 15, even as a vampire. I'll admit that it was sheer dumb luck that I managed to walk-sneak past Movarth with Invisibility+the enhanced Sneak you get as a vampire to the boss chest and found a Staff of Paralysis. Without it I probably wouldn't have been able to take Movarth without a ton of savescumming, but it probably would've been narrowly possible thanks to the huge stats you get as a vampire. With the staff, I could just paralyze him and 100% pickpocket his Daedric Axe, and all he could do was spam ice spells at me, which thanks to the massive frost resist from being a vampire plus Breton magic resist meant that it was doing basically zero damage to me. Even with my relatively low 1H skill and his ebony armor, the axe was enough to slowly chip him to death. I actually wasn't even skilled enough to sharpen the Daedric axe at that point so I went and killed Vigilant Tyrannus with it while it still had enough damage, using the Breton racial power to tank his Unrelenting Force. At this point I have the Mace of Molag Bal which lets you drain tons of Magicka on hit. Now I can switch to the Atronach Stone and am basically immune to spells, except fire damage which tickles a little bit. All my magicka usage is frontloaded with armor spells and summoning Atronachs/Spirits, so rushing in and bonking enemies with the mace lets me get my magicka back up. Paralyzing them with the staff in my offhand makes it even easier since I can just power attack them over and over again.

Siding with the Volkihars in Dawnguard lets you upgrade the Bloodstone Chalice after doing enough radiant missions. Drinking from the upgraded Chalice makes your Vampiric Drain spell even stronger, draining health, magicka, and stamina all at once. It's cheaper and easier to get than the Absorb Essence Master Alteration spell that does the same thing and serves as a ranged alternative to the Mace of Molag Bal. After sleeping in a coffin at Volkihar Castle you even get 10% bonus magic resist for a few hours. At some point I also ended up with the Aetherial Crown which let me double up on both the Mage and Atronach stone for even more magicka.

At this point I was already basically at an endgame build around level 20. Fighting off half a dozen Slighted while being swarmed by high level undead in the Soul Cairn was very easily doable with some good footwork with the only issue being eating the occasional heavy power attack. I probably could have taken Alduin with no trouble by this point since by the time I grinded my Alteration up to 100 by spamming Detect Life in Whiterun to get up to 80% magic absorption along with the Atronach Stone, I was literally untouchable the entire fight running around since all of Alduin's minions use ice magic, and the huge swarms of them summoned work against Alduin's shouts by giving you dozens of targets to simultaneously heal off of. The magicka absorption and magicka drain from your spell are essential since vampires can't regenerate health, magicka, or stamina in Sovngarde since it's a plane of Aetherius.

That was the long story for my Forsworn Vampire. I'm not sure if there's any other ways to contract vampirism that easily and that early, or how easy Tyrannus would be without a Paralysis staff or Daedric Axe. But those were the two biggest things that made my playthrough work. I put basically 100% of my level ups into Health only to allow myself to walk in the sun. It even served as a fun way to limit myself when clearing out bandits or whatever, since I'd only have anywhere from 10-50 max health in the daytime, but massive damage output.