With the new additions of Reliquary of Souls, Apotheosis, and Psyche Split, Quest Priest has gained a lot of powerful tools with the release of Ashes of Outland. Reliquary Prime has the potential to be the best card in the Quest Priest deck, if played correctly with assurance of its survival. Because it can only be targeted with hero powers and spells by you, it allows for much heavy investment into it. It can still be targeted by minions, *cough cough Flik*, so there are multiple resurrect mechanics and other strong minions that can be buffed. But your priority should be protecting Reliquary Prime.
More complex: Turn 6: Reliquary of Souls, Grave Rune, and finish with Forbidden Words
Turn 7: Reliquary of Souls, Psyche Split, and Forbidden Words one of them
Final Play: Turn 10 or later: Reliquary of Souls, Grave Rune, and Psyche Split. If you have the coin, use it to Forbidden Words one of them. If not, just run both copies into something so that both die and leave four on the board.
Besides Reliquary of Souls and Reliquary Prime, Priest received two spells that will help in the Quest Priest archetype to both complete the quest and buff minions afterwards. Apotheosis provides a reliable source of damage and healing to your hero by giving a minion +2/+3 and lifesteal.
Psyche Split is a fancy combo card to drastically increase your board strength if you already have a big minion, such as Batterhead and Reliquary Prime.
Potential changes: Grave Rune for Khartut Defender: The second grave rune may be more necessary in maintaining a board of lifesteal minions than having a singular taunt minion come back to life.