Versus a priest you fight against the number of cards he have on hand .
Starve him on cards and he won't have answers.
Let him draw and will have answer for everything.
Second problem : don't ever let his creatures wounded on the field . He will heal them. Try always to do perfect kills. If you can't kill it , better let it at full health.
Just imagine the horror scenario : you let him have 3 hurth creature on the field . He drop two
http://www.hearthpwn.com/cards/600-northshire-cleric and then he do a
http://www.hearthpwn.com/cards/671-holy-nova. He draw 6 cards. He will destroy you with those 6 cards.
Third problem : play a lot of 4 Attack creatures. He have no answer versus those , no hard removal , he got to kill them . Yeti , Azure Drake,Twilight Drake , Gadzetan Auctioner, Sunwalker , Dark Iron Dwarf , Spellbreaker etc are your best 4 atack friends. Put 3 of those on the table and he will regret it.
4th problem : take good care , turn 10 mean Mind Controll. So please don't ever play something nasty before priest turn 10 , if you don't have answer to his posible M.Ctrl. Don't play Ragnaros turn 9 if you don't have an eventually Polymorph for turn 10.