Before you can enter the new Karazhan dungeon, you need to complete an attunement quest chain. This chain start at Khadgar in Violet Citadel in Dalaran.
The quest chain requires you to complete the following four mythic dungeon in this order:
1) Halls of Valor Mythic difficulty.
2) Black Rook Hold Mythic difficult.
3) Violet Hold Mythic difficulty.
4) Vault of The Warden Mythic difficulty.
Once you have done all of these, return to Khadgar, and he will have to travel to Karazhan by using a portal. There he will give you several more quests, but all of these takes place in the surrounding area around the dungeon. Upon completing these quests, you are attuned with your account, and can enter Karazhan on all your characters.