When Liliana's Contract enters, you draw four cards and you lose 4 life. At the beginning of your upkeep, if you control four or more Demons with different names, you win the game.
2018-07-13
If the second ability does trigger, but you no longer control four Demons with different names as the ability resolves, you won't win the game.
2018-07-13
If you don't control four Demons with different names as your upkeep begins, the second ability of Liliana's Contract won't trigger. You can't take any actions during your turn before your upkeep begins.