Introduction

So, a disclaimer... I used to play Wild to D5 every season until stuff like SN1P-SN4P Warlock started happening. Not too long after, Battlegrounds and Duels came out, and I pretty much stopped playing Wild and Arena entirely. So take this decklist and guide with the GIANT caveat that I've gotten back into Wild with no star multiplier and am therefore at super-low ranks (currently Gold 10)--play the deck at your own peril if you're actually a good Wild player!

Caveats aside, the deck is super fun and not bad at all for climbing if you're at trash tiers! I've gone 24-4 for an 86% winrate climbing from Bronze 10 to Gold 10, so I figured I'd share in case anyone else is looking for a relatively easy, relaxing deck climb at those ranks!

Mulligans

Keep the quest and hard mulligan for any and all minions. Never keep spells unless you read that you can get near-immediate value (for example, Resurrect is an acceptable keep if you're on coin against aggro with Leper Gnome in your opening mulligan)--you'll draw into more than enough later.

Win Conditions

1. All your minions are "facerattle" minions ("if the face plays taunt, me still go face"), so most of the time, you can just finish off an opponent by tempoing out minions and resurrecting them.
2. Should you face a deck that has significant healing, armor gain, or damage mitigation, Xyrella, the Devout is a filthy finisher if you've had a bunch of deathrattles go off--just make sure to track exactly how many of each deathrattle minion has died so you can make sure it's enough damage.
3. Finally, completing the quest may seem like a win condition; however, the quest is in the deck not to make you win the game (your deathrattles going off does that), but rather to save you in otherwise unwinnable situations (particularly against aggro). I've found myself in quite a few situations where I would die the next turn but get saved by Amara, Warden of Hope. She won't win the game for you, but she'll usually buy you an extra turn or two for your true win condition (your deathrattles) to get there. There's usually no need to force the quest, either; it's usually complete no later than turn 7 and can then either sit in your hand as a mega-Reno or come down as a tempo play if you know the opponent has no removal.

Additional Notes

• While this is a relatively simple deck, it's not really a truly brainless deck like Pirate Warrior where you can AFK between turns. It's more along the lines of standard Mech Mage (sorry--wish I knew Wild better so I could draw a Wild comparison :/ )--it's quite a simple deck, but not a truly brainless one, since you still need to read your opponent's hand, plan your next few turns, make adjustments during the game, and maximize value. 
• Amulet of Undying is in the deck over other cards like Eternal Servitude primarily because it's cheaper and gives you a chance to cycle. If you do end up playing it, it's usually ok to just resurrect a single minion.
• When faced with a choice of minions to trade with, you'll usually prefer to trade with Kobold Sandtrooper and Leper Gnome and go face with Backstreet Leper and Shadowed Spirit. This is because Backstreet Leper and Shadowed Spirit deal more damage with their base attack than with their deathrattle. Obviously, though, this is just a general rule of thumb--there are lots of situations where you'll trade differently.
• Don't wait for uber-value when using Rally! or Amulet of Undying. If faced with the choice between playing a 2-minion Rally! or a 1-minion Amulet and doing nothing, you should almost always play the card. On a similar note, don't hold out for your 3-damage deathrattles--just take what the game gives you. If you end up on 5 mana with a Leper Gnome on board and Embalming Ritual + Grave Rune in your hand, take the 3 extra Leper Gnomes!
• The nature of the minions you run gives you unique ways both to find lethal and play around secrets when playing against secret classes. Keep an eye out for these opportunities! (For example, against a Hunter who has an Explosive Trap up, you may be able to find lethal by playing Twilight's Call before attacking face! Similarly, against a Mage who has Ice Barrier up, you may be able to find lethal by suboptimally trading rather than going face.)
• Drek'Thar is Rek'thar... especially if you get another copy with Raise Dead. (Raise Dead + Amara isn't too far off, either... it turns you into an aggro deck that has more than 100 effective life when factoring in hero powers and healing spells, as well as a ton of reach, as documented in the fun fact below...)
• Fun fact: The theoretical maximum damage off deathrattles for the deck is 258. This requires absolutely everything to line up perfectly (like getting two fully upgraded Diamond Spellstones off your Renews)... but that is the theoretical max (and doesn't factor in damage from spells, minion attacks, and Xyrella, the Devout's hero power).