I'm writing this guide under the belief that Gonk Druid is a legitimate tier 2 deck. While I don't track my stats, I've been gaining ranks in legend, a challenging task even with a meta deck, as you are matched with non-legend players, making you lose more for a loss than you gain for a win.
To compare the deck to other Combo Druid archetypes, you will find that the deck lacks ramp, aside from a single biology project. This means you'll combo later than Malygos or Togwaggle Druid. However, it makes up for that in anti-aggro tools and combo that actually kills your opponent when used, respectively.
Here's a guide to the purposes and some of the nuances of the cards in the deck, as well as when, if ever, to keep them in the mulligan. Enjoy!
Innervate: This is an Aviana Kun deck, and, while you can highroll drawing Aviana with Jepetto, two Innervates are, of course, necessary for consistency. Do not keep in the mulligan.
Biology Project: I put this in the main flex spot of the deck to help smoothen out the curve and make up for the deck's weakness of ramp. Only keep in the mulligan against control and with an already good hand.
Lesser Jasper Spellstone: Despite being only one mana, this is one of druid's strongest anti-aggro tools and a great general removal spell. Keep in the mulligan against aggro, or broadly, classes with early, aggressive minions.
Secure the Deck: This card pushes your Gonk combos over the top. While there are a variety of ways you could use this card, I find it best to play prior to the combo turn and to swing once beforehand, giving me three Claw spells after my first attack that I can use to clear the rest of the board and swing face with. The armor it grants is just as important, though, as its gives you the stability to get enough swings in to OTK an opposing aggro deck. Although it is often played on turn 1, never keep Secure the Deck in the mulligan.
Feral Rage: This is the second and inferior attack buffer, but it is not to be underestimated. The four attack it grants ensures OTKs and it has versatility to be used to gain 8 armor before or during the combo. Only keep it against Mage, as the armor it grants can be invaluable against Secret or Quest Mage.
Ferocious Howl: This is a must in any nonaggressive Druid deck. It is great draw, great armor gain, and consistently upgrades your spellstones. Always keep in the mulligan.
Branching Paths: Branching Paths is a more versatile, but inferior Ferocious Howl. Still, the draw and armor gain it represents makes it a must in the same deck Ferocious Howl is in. Always keep in the mulligan.
Juicy Psychmelon: While Gonk titles this archetype, Sathrovar defines it and Aviana and Kun enable it, Juicy Psychmelon puts everything together in a nice consistent package. And it draws Jepetto, too! A fringe use is, after copying Gonk with Sathrovar, you could use Psychmelon to draw the copy in your deck and play it for a fourth Gonk, but there are usually better things to do with your mana. Keep in the mulligan against control, combo, and/or with the coin.
Oaken Summons: The Oaken Summons package is instrumental in holding off aggro. It alone can't win games, but the armor gain and tempo swing it creates is priceless. Always keep in mulligan.
Poison Seeds: Along with being one of the strongest removal spells in the game, putting a board under control and often preventing your opponent from summoning a minion, Poison seeds works in this deck to set up an easy board to combo off of. Saving one until before you combo against a deck with big minions and especially big taunts can be a useful strategy. Keep in the mulligan only against Paladin.
Spreading Plague: The iconic stalling spell is, of course, an auto-include in the deck. Spreading Plague the turn after Poison Seeds can be devastating, but, at the very least, the card can give you one more turn to assemble your combo. But be wary of Devolve and the Warglaives of Azzinoth. Keep only against Warrior, Paladin, and Rogue if you have a decent curve leading up to turn 6.
Gonk the Raptor: Self explanatory. Never keep in mulligan.
Malfurion the Pestilent: Aside from being a great anti-aggro card, the attack buff granted by his hero power can be very useful. Only keep in mulligan against Mage.
Aviana: Self explanatory. Never keep in mulligan.
Kun: Self explanatory. Never keep in mulligan.
Archmage Vargoth: The minion that makes Oaken Summons truly good. With two Ritualists, you'll rarely recruit Vargoth, but it is surely a swing when you do, and the card can be flexibly used with your other spells, often Ferocious Howl and Branching Paths, to get loads of armor and draw. Just make sure you don't mill anything if you're looking for a card. Never keep in mulligan.
Zul'Drak Ritualist: The main purpose is to be recruited via Oaken Summons, but the Ritualists can be saved against non-board-centric decks to create minions to attack into. Never keep in mulligan.
Jepetto Joybuzz: One of the worst cards in the deck, and yet it's a swiss army knife. Missing Aviana or Kun? Fetch 'em. Missing a Ritualist? Fetch it. Need a dummy for explosive runes or to be ratted out? He can do it all. And drawing Aviana with Jepetto gives you the potential to combo even faster than Mecha'thun Warlock with Hemet on 6. What pushes it over the top, though, is that the card is a free draw off Psychmelon. Never keep in mulligan.
Sathrovarr: If you want to copy minions in an Aviana Kun dead, Sathrovarr's the guy to do it. Not only does he give you an immediate copy and a 1 mana copy (and one in your deck in case you need it), but he is tutored by Psychmelon along with the rest of your combo. 95% of the time, you should use him with Gonk, but there are fringe scenarios when you don't have Gonk or you lack the durability to combo when you want to copy Kun instead. After playing Aviana, Innervate, and Kun, play Sathrovarr on Kun adding a 1 mana copy to your hand that can be used to gain 10 armor. Using Jepetto and/or Psychmelon, you can fetch the second copy to gain 10 more. Often, you should focus on assembling the standard combo against aggro, but keep this in mind as, if you don't die the turn afterwards, the tempo swing is game-winning. Never keep in mulligan.