I got this idea from a YT video, from the channel Bad Boy Gaming. Here's the video: https://www.youtube.com/watch?v=7SPS-mzB7g4&t=380s
The deck from the video has the possibility to kill at turn 1, but it is not Modern legal. I've tried to modify it and make it fit into this format, and I've managed to create a deck that is able to kill at turn 2! And for just 20$!
The main way to kill with this deck is a combo that consists of 2 Myr Retrievers, Altar of the Brood and Heartless summoning. To increase the probability of the Turn 2 kill, I've added Myr Moonvessel. Here's how it works:
Turn 1: We play a Swamp and Myr Moonvessel (or Altar of the Brood)
Turn 2: We play a Swamp and we play Heartless summoning. If we played Myr Moonvessel, it dies because of the -1/-1, so it gives us an extra mana, with which we play Altar of the Brood in case we hadn't. Now, since "creature spells you cast cost 2 less to cast", Myr Retrievers are free to cast. We play a Myr Retriever, but when it hits the battlefield, it instantly dies because of the -1/-1. We play another Myr retriever, and it dies again, but while it dies, the other Myr Retriever comes back from the Graveyard to the Battlefield. And this happens again, and again, and again, infinite times.
Every time this one of the Myr Retrievers hits the battlefield, before dying again, it triggers Altar of the Brood, making the opponent "put the top card from his or her library into his or her graveyard".
So, basically, we'll empty our opponent's library, and he or she will die during their next upkeep, since they are not able to draw from their library.
The other cards are there in case we don't get the full combo from the beginning.
-Diabolic Tutor is there, obviously, to search the cards we need to complete the combo. There is not a lot of search in this deck, but all cards are useful and the probability to get the right cards is good, so it won't be a problem. In case we needed more search cards, we could include Shred Memory (which is in the Sideboard), because its Transmute lets us look for Myr Retriever and Heartless Summoning.
As Bad Boy Gaming says in his video, you can add Beseech the Queen istead of this, but it is considerably more expensive.
-Dimir Machinations is there to start milling the opponent to pump up Wight of Precinct Six, which will be our main creature while we try to get the full combo. We can also use Dimir Machinations to dig into our library and get the deadly combo.
-Doom Blade, Tragic Slip and Gatekeeper of Malakir are there to destroy annoying creatures that can put us in a difficult situation while we're looking for the right cards, and at the same time, to pump up our friend Wight of Precinct Six.
-Wight of Precinct Six, as said before, will be our secondary tool to attack and defend while the combo comes to our hands.
-Aphotic Wisps (and Nighthaze, at the Sideboard) are there to make Wight of Precinct Six a little bit better while letting us draw an extra card to get the combo.
In conclusion, the ideal starting hand would be:
-2x Myr Retriever
-2x Land
-Altar of the Brood
-Heartless summoning
-Myr Moonvessel
(Taking into account that, if we don't start, we get an extra card, and yet another one during our Turn 2 upkeep).