I updated the deck with some K&C cards. As always, suggestions are appreciated.
This deck is not Tier 1 or 2, just one that'll help you climb the ladder a bit near rank 15 or lower, and can be really good if things turn out properly. It's meant as a solid, fun, casual deck rather than one for reaching legend. Have fun with it!
This is an aggressive priest deck that focuses on a few "at the end of your turn" synergies to create a powerful early game threat with minions that buff each others. It has also got a significant late-game tool, Shadowreaper Anduin, that can help you survive and finish off low health enemies.
Here's a video with a few pretty decent matches so you can see how the deck works, and a replay posted in the comment section by _barat_ :)
The Game Stages:
The best cards to mulligan for, as a turn 1 drop is Northshire Cleric. You'll also want to get some 2 or 3-drops, such as Fallen Sun Cleric or Shadow Ascendant. Shadow Word: Pain and Happy Ghoul are both very good cards to keep if you are encountering many aggressive decks.
Needless to mention always replace your expensive drops, such as Shadowreaper Anduin and Cobalt Scalebane, but also Mind Blast, which you are in no rush to play, as it only serves for applying pressure or reseting your Void Form Hero Power in the late game.
In turn 1, the best possible drop is Northshire Cleric. It has good stats and can be a good defense against early threats, and it's unlikely it'll die by turn 2, in which you can buff it. If you don't draw it though, you can play Acherus Veteran if your opponent has not played any minions and is not a Mage, Druid or Priest (which have easy removal with their hero powers), or wait until turn 2 to drop the Radiant Elemental or another 2-drop.
In the following turns you'll want to play Shadow Ascendant and Master Swordsmith, together with Drakkari Enchanter and Mirage Caller to maximize their value, and focus on keeping them alive by buffing them with Power Word: Shield, Fallen Sun Cleric or Kabal Talonpriest.
As mentioned previously, Shadow Word: Pain and Happy Ghoul are your only defenses against early threats, so you'll want to create a more threatening board than your opponent in order to win the early game.
There are two possible scenarios in which you can find yourself upon reaching turn 5 or 6. If you have been able to keep the threat on board during the early turns, you'll have a lot of very powerful minions, that even powerful removal tools like Flamestrike can't kill. In that case you have probably won already, Each turn you'll deal more and more damage, and your minions will become harder to deal with.
However, you can also be in a less favorable position, in which your opponent has been able to deal with your minions, and you have little or no board presence. In that case you'll have to keep yourself alive by playing small minions, drawing cards and healing yourself with your hero power until you reach turn 8 and can drop Shadowreaper Anduin.
Your best tools to keep up the pressure in the mid game are Cobalt Scalebanes, which use the same synergies as Master Swordsmith and Shadow Ascendant, and are very good to buff your minions, aside from being a strong body on board. Prince Valanar can also protect you and help reach the late stages of the game.
If you happen to reach the late game, there is only one card that can help you win the game, and that is the so-mentioned Shadowreaper Anduin. Upon playing him, you'll destroy all "big" minions of your opponent if there are any, and your own, if you don't mind the sacrifice. He'll also grant you 5 armor, which can be essential or just useful for survival.
The hero power can finish off the remaining minions, and then start hitting your opponent's face. Unlike other aggressive decks, you can benefit from drawing cheap minions or spells, as they reset your hero power. The two copies of Mind Blast deal a significant 10 damage in total, and can deal 16 damage in a single turn if alternated with your hero power.
Overall this is an aggro deck, and you won't want to reach the late turns of the game, but if you do you have Shadowreaper Anduin to scratch off those points of damage you weren't able to deal during the early turns, or destroy your opponent's threats and begin flooding the board with minions again.
This is a list of some of the changes that can be made depending on the different decks you're facing, your own playstyle, or if you happen not to have some of the cards, especially the epics or legendaries:
I will update this section with suggestions from the comments section, as I can't try all different possibilities and don't have all the cards myself.