This is what I plan to run as priest when I can get to this point. Currently been playing around with a somewhat weaker version with mostly decent success. Here is the run down for the choices made:
Holy Smite: Good in the early game especially if you will meet up against highly aggressive decks. As a cheap spell, it can still see good use in the late game as well.
Power Word: Shield: Useful in the early and mid game also allowing you good card draw potential in conjunction with Northshire Clerics. Only ran one as it will be less likely used for the card draw since the clerics will fit that purpose well enough.
Shadow Word: Pain: Obvious why it is in, but still good to keep in mind of the deck your against so that you can get the best targets possible.
Shadow Word: Death: The "Oh Shit" card. Any minions that could give you trouble in the early game such as giants/buffed creatures out of the way before they can do any real damage. Late game for its cost gives you great board presence with your late game minions.
Holy Nova: Will help to set you up for the late game as well as being incredibly powerful when used alongside Prophet Velen.
Holy Fire: Has a high cost, but enough damage to deal with majority of minions you will come across while also giving a bit more stall time with health. Can also act as a hero nuke when used with Prophet Velen to deal potentially 20 maximum damage.
Mind Control: Can not be a bread winner without the bread to your butter. Easily the most powerful card in the game which alone can make your opponent wary of playing certain minions. Be sure when playing against another priest that you are not the aggressor with this card.
Northshire Cleric: Two of the few low cost minions you will be able to make use of and likely the most important. The card draw potential is necessary as the early game will be the weakest point for this deck.
Lightwarden: Can be useful in the early game and late game. Used alongside northshire cleric will make it easier to reach the late game.
Mana Wraith: If you are lucky to get this card at the beginning, will allow you to stall for a turn or two as well as giving you the ability to decide when to sacrifice it for continued board control mid game. Northshire cleric and lightwarden also have the capability to be played on turn two if you coined this minion out.
Injured Blademaster: In a priests hand, it becomes the best normal 3 cost drop in the game. Works well with the northshire clerics and lightwarden. Also capable as a 5 drop.
Chillwind Yeti: Their are actually quite a few different choices you can go for in place of this. Yeti just seems like the best since it is a 4 attack minion with above average health.
Sylvanas Windrunner: Fantastic control card that will either force your opponent to kill off all his minions or risk you taking one they would rather you not have. If you do not feel comfortable with this card, then consider running Bloodmage Thalnos to function alongside holy nova.
Argent Commander: Charge with divine shield.... enough said. Can consider swapping sunwalker for another commander.
Sunwalker: Taunt with divine shield plus 5 health.... refer to above. With no silence, this card will generally trade (at minimum) 2 of your opponents cards. Can consider swapping commander for another sunwalker.
Black Knight: If any card will get on your opponents nerves, it is the Black Knight. Quite often you will see players run Sunwalker in constructed to which this is a solid answer to that if not more. Tirion beware.
Cairne Bloodhoof: This card can give your opponent a tough time as it is an improved version of the harvest golem. Can trade incredibly well without fear of it dieing the first time around. Practically has the health of the druid's ancient of war.
Prophet Velen: Mostly gonna be used for the improved healing capability to keep your strong minions alive. Does have impressive functionality with certain spells though. If anything he will become fodder to set up for Ragnaros or Ysera.
Ragnaros the Firelord: This may play with rng, but it has too many positive points for you to care. When it deals damage, it does not take damage and can bypass taunts that would otherwise cause you trouble. If somebody is foolish enough to silence him, then you have control of an 8/8 minion. If you really are not into the rng element, you can consider swapping him out for Gruul.
Ysera: Best legendary in the game if not the best minion in game. Not only is it a 4/12 minion, but it also gives you some very powerful cards at the end of each of your turns.
Mountain Giant: Has plenty of options in how you can play it. Since you will lack in the early game, but can run northshire clerics. This minion can be a saving grace to buffering the early and mid game.
The choices made are very sound, but like any other deck it has weaknesses. Primarily with so many mid to late game cards, a lot of games you could end up exposed in the early game especially against very aggressive decks. Another weakness is that you do not run silences nor oozes. It would not matter to run oozes as it is useless unless you were against a warrior that uses gorehowl since you will have minimal early game to save. Despite these factors, if you are able to stall into the late game, you will win majority of the time.
It has some truth though if you come to a point for playing him before you have enough mana for rag and ysera. He is a powerful minion, so players will likely unload w/e responses they have to get rid of him thus making it more difficult to deal with Ysera and Ragnaros. Also helps that once you get to 6 mana, you have a lot of plays that become difficult for the opponent to deal with.