You can trigger this quest by visitng the alchemy shop " The White Phial " in Windhelm . Here, you will here the shop owner Nurelion and his assistant Quintus arguing. Ask him about the argument and then about the White Phial and you will offer to help him find it, starting this quest.
You can ask him more about the phial after taking the quest to learn that it is indeed a rare and valuable object, able to replenish any liquid put into it AUTOMATICALLY.
Neurlion will say that the Phial was buried with its owner, and you’d have to be a master alchemist to get to it, but he has prepared a special mixture to help you. You’ll get Nurelion’s Mixture now. Head outside and open the map to see the quest marker takes us to the Forsaken Cav e, west of Windhelm but before Nightingale Inn. Head on over there now using whatever means you can and enter the cave’s opening.
At the entrance of the cave will be a couple of Snow Bears. Dangerous if you aren’t ready, and a good measure on whether or not you can handle yourself well enough. Kill them both and head down a spiral staircase to a tomb area. As you may have guessed, Draugr inhabit this area, so be ready for them and get in some sneak attacks/first hits when you can.
Follow the path and loot the chest, then take out 2 Draugr and continue to a burial area. Remember, the Draugr can come OUT of the wall here, so get in the first hit! There are two more Draugr in the burial area to kill and a door nearby that you can lockpick to access a chest.
Loot the place and continue on down the linear path down to an area where all the dead lay horizontally. Watch out for enemies on the slabs here as well as patrolling Draugr. There is easily 6+ Draugr in this area, as well as fire traps on the floor. Be sure to check the embalming tables here too for gold bars (I found two of them!). Continue down a hallway watching out for floor traps and more goodies on the bookshelves until you can enter the Forsaken Crypt.
Head into the crypt and kill the Draugr. Ransack the shelves and follow th path up and around to a higher level, killing another Draugr along the way. Keep following the path killing Draugr and looting until you get to a bigger room with an iron door on the left and some Draugr. Take out the Draugr and pick the door if you wish as there’s a chest behind it. Don’t step into the circle in the room though, as it’s a trap.
Keep heading down the path until you can open another door. Up ahead past some swinging-blade traps is a coffin straight ahead of you. Get ready for a fight (I’d highly recommend a companion here and perhaps a powerful shout like slow time or something for crowd management).
As you enter the room, a DEATHLORD will wake up from the main coffin with FOUR other Draugr of various strengths appearing from the sides of the room. Take them all out. There is actually a WORD WALL that teaches a MARKED FOR DEATH word in this room, but it’s got two staircases leading up to it that I like to use to separate myself from the many Draugr.
After the fight get the new word of power and loot the chest, then head under the word wall to find an "Ancient Bowl". With the mixture that you received, you can activate the bowl and open up a secret room beyond. In here you can find a TON of alchemy items, an alchemy lab, an ALCHEMY SKILLBOOK named "A Game at Dinner" on a shelf, and of course the White Phial on a pedestal. This phial is cracked, though, just take note of that.
Ok, now it’s time to exit (use the iron door to the left of the word wall for a quick exit) and return to Nurelion. Once you show him the vial, he’ll acuse you of breaking it. In the end, he’ll only give you FIVE septims and take the phial (we will then, interestingly enough, go lay down on his bed and when you interact with him it will only say he is asleep and dying).
Go speak to Quintus now who will have taken over running the shop and will give you a more fitting reward of 500 gold septims. He will then head off to make Nurelion’s last days more bearable. And with that the quest will be complete.
Note: After 3-4 days, you will be contacted by a courier who will give you a letter from Quintus. It appears that he has found a way to REPAIR the White Phial, and wants to talk to you about it. This starts the quest " Repairing the Phial ", so look it up if you need help with it!