Hearthpwn

Current Revision 4/28/16:
Brann definitely feels too slow, replaced him with another shifting shade, Black Knight can be teched out for either another death or Harrison Jones.  You can also replace a Blackwing Corruptor for another death or excavated evil, depending on your current matchups.

Information:  
Dragon Priest is a mid-range deck that exploits dragon synergy in order to play out strong mid-range minions on curve. Dragon Priest's game plan is to play out minions on curve and heal them in order to get as much value out of them as possible, saturating the opponent out of answers. The power of the deck lies in the amount of value it can generate through the dragon package available in the deck. Priest's Lesser Heal suits the mid- to late game approach of dragon decks as well, making Priest one of the most powerful classes with a dragon archetype.

Mulligan: 
While there are specific mulligans for certain classes, generally you mulligan for all of your early game. Namely, Wyrmrest Agent, Northshire Cleric, and Blackwing Technician. Also, do not mulligan for Museum Curator unless you know the match up well and have a decent starting hard.  A lot of the newer deathrattle minions aren't as strong in Standard.  

Card Choices/Replacement: 

Harrison Jones - He was previously on the list, I just started running into more taunts and teched him out for The Black Knight.  Choose the one you're running into more often.

The Black Knight - This is just a tech card for now while many people are still experimenting with C'thun decks. with all of the C'thun decks running taunts like Crazed Worshipper, Dark Arakkoa and Twin Emperor Vek'lor this card will help you gain back a strong board state without buffing their c'thun or losing too much tempo.  

Sylvanas Windrunner - Run another death if you don't own her or feel she's too slow.

Shifting Shade - It acts as another strong body that can hide behind your taunts, but also gives you a form of card draw from the opponent. Decks right now are running lots of big minions, impactful small minions and spells.  Being able to get 2 extra cards from their decks help a lot more than you would think.  

Brann Bronzebeard - I generally feel like this card is a little too slow but if he can be kept alive, he'll help you win more. You can also replace this card with Elise Starseeker for a more late game focused deck.  If you want to play both of these cards, drop a Shifting Shade or The Black Knight.

Deathwing, Dragonlord - Big body, strong stats, great deathrattle. This guy can be your finisher with style.  Try to play him around or after their big removals. Most decks still aren't running silences, but he may not be an option later when the meta is more refined.   You could also try playing Deathwing, with a lot of the voluminous board states with big stats, he can become another board clear that this deck is clearly still missing.

Blackwing Corruptor - Sometimes he doesn't feel like he does enough for his mana cost, he has a hard time dealing with lots of small minions, doesn't kill many of the newer minions and doesn't fare well against spell decks. If you're having trouble with aggro or control you can tech 1 of the corruptors out for a Excavated Evil or Cabal Shadow Priest. If you want more late game focus, tech it out for a Justicar Trueheart.    

State of the deck within the new meta: In progress..

Match-ups: In Progress..

Final Thoughts: 
I spent roughly 12-14 hours playing this deck on ladder from rank 9 to Legend yesterday and will continue today. Tweaking and teching for the current meta, if you have any suggestions please feel free to let me know. I'm currently at roughly a 65% win rate.  As I get a larger sample size, I'll fill in match-up details and the state of this deck in the new meta.