The idea behind the deck is to generate mana by milling yourself down to almost 0 cards, and then use elixir of immortality to repeat the process until you have the mana to cast Amulet of Vigor > Door to Nothingness > DtN's ability. Simple concept; complex execution.
How does the deck operate exactly? The only card that makes it tick is the lovely Vedalkin Archmage. In case you didn't read the cards in depth, he reads, "Whenever you play an artifact spell, draw a card." With this ability and the 18 0-drop artifacts (+3 1 drop artifacts) in this deck, there's potential to draw an INSANE amount of cards! For every artifact you draw, you may draw another card, and for every retract you draw, you may draw as many cards as you have artifacts for 1 blue mana! If you consider the fact that retracting all your artifacts back to your hand only requires 1 mana, and retracting will allow you to play your mox diamond for a 2nd time and generate another mana with it, Retract is virtually free!
Once you manage to draw your entire deck, you can use elixir of immortality to shuffle it + your graveyard back into your library. At this point, your graveyard will most likely consist of 3 mox diamonds (you have to sacrifice 1 mox diamond to play another due to the fact that they're legendary), your 2 Rite of Flames, a couple Mishra's Baubles, some Retracts, and some sift cards. With the artifacts/ retract you have sitting in your hand, you can draw your deck again, playing your Elixir of Immortality, 3 Mox Diamonds and 2 Rite of Flames. Repeat this process as many times as necessary, using the mana from the Rite of Flames to play/ sac the elixir, and using the mana from the Mox Diamons to generate whatever color mana is required to repeatedly play retract and eventually play/ use Amulet of Vigor and Door to Nothingness.
P.S. - Note that if your elixir happens to be shuffled to the bottom of your library, you will have to play you 2nd elixir that will be sitting in your hand due to the fact that playing an artifact with no cards left in your library will cause you to lose the game.
Ideally, you want to keep a hand with 2+ mana sources (even if you can't use them at the start), some card sift, and a Vedalken Archmage. If you don't start with the Archmage, that's fine. You will be able to sift down to the Archmage before you're beaten down to 8 or less life in most games.
Turn 1:
On turn 1, you want to play a scry-land if you started with one, and scry anything that isn't ABSOLUTELY vital for the your combo. Then, use any Gitaxian Probes that you have in your hand.
Turn 2+:
Once you have a mana source, use any 1-drop draw cards that you have. In some cases, you may want to play some artifacts before playing a Vedalken Archmage in order to make Thoughtcast affordable. It depends on how many artifacts you have left in your hand and how rushed you are to get your Vedalkin Archmage. Once you have your archmage along with the mana to play him, decide whether it's likely you'll win the turn you play him. If it is, go ahead and cast him and attempt to draw your entire deck THAT TURN, or at least draw to another archmage to play the next turn if they kill him. If you can't win that turn or draw to another archmage, hold off until you can.
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At first glance, this deck seems impossibly complex to pull off, and it may seem like it will almost never work due to the very specific cards needed for its combo. A friend and I HAVE actually tested it (quite extensively (; ), and it has a much higher success rate than it seems like it should! It's by no means a top tier, competitive deck due to the fact that its combo will sometimes fizzle, and also many decks in modern are turn 3-4 wins while this deck is a turn 4-6 win. It's intended to be a funny deck that makes your opponent's jaw drop and scream "WHAT THE F**K WAS THAT!" when they see it go off for the first time! And it does work consistently enough to be a viable deck in a less competitive, Friday-Night-Magic-type atmosphere.