Version 2.0 Update! After some playtesting myself and gathering feedback from those of you in the comments, I've targeted the following weaknesses of the initial 1.0 version: how the deck struggled properly contest the board, and could not easily clear larger threats. To do this, I have removed 7 cards: 2x Feral Rage, 2x Pathmaker, 2x Dire Frostwolf, and 1x Rustrot Viper. In their place, I have added the following:
2x Vibrant Squirrel: A decent turn 1 play that can fill out your mana curve in the midgame, and combos with all best synergies. The acorns extend the benefits of Heart of the Wild's wide buff aura, but even more importantly can be used to clear minions with the newly added Animated Broomstick.
1x Lunar Eclipse: Useful removal at any stage, this pairs well with the numerous 2 mana spells, and can pair with Eye of C'Thun and Heart of C'Thun. (Only reason I don't have 2 is limited deck space). If you are having trouble with Warlocks, you may consider putting Rustrot Viper back in instead.
Goal: You want to use beasts to aggressively contest the board and smack your opponent in the face, while passively assembling C'thun, the Shattered.
Mulligan and Strategy Against Aggro & Quest Warrior/Warlock & Any Paladin
You want Druid of the Reef, Vibrant Squirrel, Clawfury Adept, Shan'do Wildclaw, and potentially Heart of the Wild. Choose the 1/3 Taunt turtle form for the extra HP to survive long enough for Heart of the Wild. Jerry Rigg Carpenter is not the best card to keep but is still an OK turn 2 play. Don't ever keep Wildheart Guff in your opening hand, you can draw him passively.
Mulligan and Strategy Against Control & Quest Mage & Phylactery Warlock
Vibrant Squirrel, Jerry Rigg Carpenter, Capture Coldtooth Mine, Shan'do Wildclaw, and Wildheart Guff are your best cards to start with. You want to get Acorns shuffled and C'Thun pieces drawn as soon as possible (you also don't want to draw the squirrel later in the game, best to play them early).
Outside of those C'Thun Pieces, you want to use your resources wisely. For example, only play Venomous Scorpid when you are sure it will kill something big, so waiting to combo with an Animated Broomstick is ideal (the poisonous is often more important than the spell it generates). Save at least 1 copy of Moonlit Guidance until after C'Thun, the Shattered is assembled to try and generate a copy into your hand, this works especially well with Solar Eclipse to double your chances on getting what you need.
The deck has Wildheart Guff and the 4 C'Thun pieces as the highest costing cards at 5 Mana, meaning they can be reliably drawn with Capture Coldtooth Mine. Wildheart Guff also helps accelerate your card draw per turn with the new Hero Power, not to mention the +5 armor. I have found it is actually best to play Guff when you have 6 available mana rather than 5, because you gain a full mana crystal when played, leaving you with 2 mana to work with. At the very least, you can Hero Power with that 2 mana.
Jerry Rig Carpenter can draw and split Capture Coldtooth Mine. If played on Turn 2, you can in a pinch use lower costing draw to try and find either Druid of the Reef [/card] or [card]Vibrant Squirrel . If played on Turn 4, use the higher cost draw to find and play a 5 mana card next turn.
Moonlit Guidance is best used either at 2 Mana to find either Shan'do Wildclaw or Heart of the Wild (only if you played a Beast Turn 1), use at 7 mana to draw 2 copies of any card in the deck, or at any point after C'Thun, the Shattered has been assembled.
Problems You May Encounter
The armor gain of Deathrattle Priest may pose a problem if you are unable to kill with 30 damage. Even if you generate a copy of C'Thun, the Shattred, they can still play the Xyrella Hero card. Not impossible to beat, just hard.
Phylactery Warlock can potentially assemble their combo faster than you do, so you will need to assemble C'Thun, the Shattred as fast as possible. The aggro strategy is less likely to work because they have so much removal, but you can still try that route too.
Quest Warrior is unfair, try to beat them down with beasts before they kill you.
Against Paladin, save your Venomous Scorpid [/card]+ A[card]nimated Broomstick for the best removal moment.
I hope that these changes help better your experience playing this deck style, best of luck!