Control Priest 3.0

Aggro Mulligans

These cards help you out-tempo your aggro opponents.  The northshire cleric is a good threat to keep them considering the costs for playing small early drops for tempo.  Geode will also encourage them to remove and/or avoid playing smaller drops.  

Potion of Madness will simply out-tempo them if they go in aggressively and pain/talonpriest/soulfiend are just great value.

Midrange Mulligans

Against midrange decks you will want to keep cards like Shadow Word: Pain to remove high-value minions and Shadow Word: Death to remove their big-tempo-and-stats minions.  Fel Orc Soulfiend is a solid body to play in the early to keep them from getting out of control for board.  You don't need it to live long.  Just long enough to buy you some time.

And in case things get out of hand, keep Dragonfire Potion to clean up the mess and regain the tempo.

Combo Mulligans

Against Combo Decks you want to mulligan EVERYTHING to get Dirty Rat and Kabal Courier.

You can either choose to try to rush down the combo deck, but in the case that should fail you need your security line: Dirty Rat.  If you can play Dirty Rat into the right hand you can completely disrupt their combo, allowing you to win the game.

Alternatively you can play Kabal Courier (preferably with Brann Bronzebeard) to search for cards to disrupt their combo.  You can look for an Iceblock, an Ice Barrier, Potion of Polymorph, Mayor Noggenfogger, a taunt... Find SOMETHING to counter your opponent's combo.  

Control Mulligans

Against control you want to do what you can to gain the card advantage.  If you can get the card advantage you will most likely win the game.