In the mulligan, you should look for Firetree Witchdoctor (so you have something to help with your opponent's early board), Northshire Cleric (for draw), Twilight Drake (a great turn 4 drop). Against aggro, I'd look for either a Duskbreaker or a Tar Creeper instead of the Drake.
Your early game is all about drawing with the Cleric and maybe a Power Word: Shield if you get one. The spell from the Witchdoctor can help with early minions.
By the mid game, you should drop your Creeper or Duskbreaker (especially Duskbreaker if you haven't been able to kill off their minions) for aggro or the Drake for for control.
Later on, you'll have access to Psychic Scream and Shadowreaper Anduin to help clear wide boards and big minions. The big finish is Alexstrasza and your Mind Blasts. Using Shadow Visions to find more copies of Mind Blast will seal the deal. You can also use Shadow Visions to find another Holy Fire if need be.
Don't worry about some early hits, there is enough healing in the deck. In the early game, use your hero power to heal the Cleric and draw those cards. You'll be able to starve your opponent later with Primordial Drake.
It's going to be a long game that comes down to Alex and your Blasts. If they can heal or armor up after you play her, you've likely lost your shot. Anduin's DK hero power will be especially useful here, as you can get that extra damage in between the Blasts.
Cabal Shadow Priest and Twilight Acolyte work together to deal with big threats, Duskbreaker is a great board wipe, and the Witchdoctor is versatile, but all need dragons in hand, so be sure to hang onto one.
The Omega Medic should only be played after 10 mana, you will likely need the healing and its not going to do much work on the board beforehand.
A couple people have asked about replacements. If you're running into a lot of weapons, slip an Acidic Swamp Ooze in there, and if you need another small minion, try Faerie Dragon. If you don't have Primordial Drake, try Amani War Bear.