Graydle Kaoju

First of all the field spell Kyouto Waterfront will come in handy for searching the Kaijus and using their effects, you'll be generating Kaiju counters regularly so Kaiju Files and Kaiju Capture Mission are not needed though you can use them if you wish.

First lets get to the main stategy is to summon  a Kaiju on your opponents side of the field, summon a Graydle, destroy it either through battle or through a card effect of a certain type and take control of it, If you don't have a Graydle in hand summoning your own Kaiju will do fine as well.

Each of the 3 main Graydles can be destroyed in ways other than battle.

Graydle Eagle can be destroy by a monster effect, preferable with Graydle Slime, allowing you to take control of it. If you set this card on the field you opponent will most likely try to destroy it with a monster deck especially if they don't know what kind of deck you're playing, this will disrupt their plays unless they have a way to negate it.

Graydle Alligator can be destroyed through a spell effect preferably with Graydle Impact or Interrupted Kaiju Slumber. If you use interrupted while the field spell is active summon Gameciel on your side of the field and any other Kaiju on your opponents side, Graydle Alligators effect will active allowing you to take control of your opponent's Kaiju and it will immediately be destroyed, this will generate 4 Kaiju counters allowing you at least 2 negates.

Graydle Cobra can be destroyed though trap effects which is where Graydle Split comes in, unless you opponent is running battle traps like Mirror Force which even then they most likely wont activate unless you have another monster on the field you most likely will be destroying it through battle.

Graydle Slime & Slime Jr. are your tuners so they're required to make synchro plays.

Graydle Slime can be special summoned by destroying 2 Graydle cards you control, this includes spells & traps so you can destroy Gradle monsters equiped to monsters you control, after that you can special summon a Graydle from you graveyard to set up a synchro summon. If you destroyed Graydle Eagle while it was in the main monster zone you can take control of a monster your opponent controls as well to use yourself as well as clear their field.

Slime Jr. can special summon a Graydle from your graveyard and then one from the hand of the same level, though be aware that if you activate his effect you will only be able to summon Graydle Dragon & White Aura Whale though synchro summoning for the rest of the turn.

For the Kaijus they each have useful effects if you have the field spell out.

Dogoran can clear your opponents field of all their monsters at the cost of 3 counters, Gandarla can halve the ATK of all other monsters on the field on a quick effect for 3 counters, Gamaciel gets a non once per turn negate for 2 counters, & Jizukiru can negate any effect that targets one card on the field and destroy it on a non one per turn quick effect, be aware that your opponent can use these against you. Mecha-Dogoran can remove 2 counters once per turn and equip a Kaiju from your hand or graveyard and gain ATK equal to that monster, I pretty much only have it in this deck so I can have another level 8 for synchro plays, so you can replace this one if you want.

For the spells Graydle Impact is useful for destroying your opponents cards or search out Gradyle card during the end phase. You can also use on Gradyle Alligator to make your opponent go -2 if the have a monster.

Interrupted Kaiju slumber will clear the field of monsters, special summon 2 Kaijus on both sides of the field and force them to battle. As mentioned before, if used in combination Graydle Alligator, you will clear your opponent's field of monsters. You can also use it's graveyard effect during your next turn to search another Kaiju.

Lastly Terraforming will help you search the field spell.

For the traps Gradyle Split will boost the ATK of on monster you control by 500 but it's main purpose is destroying one of your monsters and then summoning 2 Gradles from your deck so you can synchro summon. If used with Graydle Cobra you can take control of one of you opponens monsters as well.

And finally the trap Graydle Parasite will summon a Gradyle from your deck during your opponents battle plase and make them think twice before attacking. during your battle phase if your opponent has no monster ,but there's a monster in their graveyard that you want for your self, you can special summon it to their field and then crash into it with a Graydle to take control of it.

Now for the synchros, the main boss monster of the deck is Graydle Dragon, if summoned you can target and destroy cards your opponent controls up to the number of Water monsters used to summon it, which will be 2 if you used Gradyle Slime & 3 if you use Slime Jr. If it's destroy you can summon a Water monster from the Graveyard with it's effect negated which won't matter for any monter in this deck aside from White Aura Whale's piercing damage.

Speaking of White Aura Whale it has a non cost Lightning Vortex effect when synchro summon & can make 2 attacks on your opponents monsters with piercing damage. It's other effect will cause it to revive itself as a synchro turner by banishing a Water monster from the grave if it's destroyed by your opponents cards, which is where the Kaijus come in. If Gandarla is on your side of the field and you have enough counters, I recommend crashing White Aura Whale into a monster with higher ATK and chaining Gandarla's effect to it's revival effect. This effect will help summon 2 other synchro monsters.

Leo can be summoned using a level 8 monster and Slime Jr. Your opponent won't be able to target it outside your main phase 2 and will have a hard time beating over it due to having 100 more ATK than average for a level 10.

Star Eater can be summon by using a revived White Aura Whale and a level 3 Graydle, Your opponent won't be able to negate it summon with something like Solemn Judgement. It will also be unaffected by your opponents card effects until the end of the Damage Step  while it's attacking.

Final Sigma can be summoned using Jizukiru & Slime Jr., It is unaffected by card effects outside it's archetype while in the extra monster zone which will protect it from Interrupted Kaiju Slumber. It also does double battle damage while in the extra monster zone, so I would not reccomend summoning more than one.

Finally the strongest monster in this deck is Phantasmal Lord Ultimitl Bishbaalkin, you can call it an bonus boss monster. This can be special summoned using a revived White Aura Whale and a level 8 monster. It can not destroyed by card effects and can summon the same number of tokens on both sides of the field on a spell speed 2 effect gaining 1000 ATK for each monster on the field and easily going up to 10000. You can summon a Kaiju on the opponents side to ATK for game.

That's my version of the Graydle Kaiju deck, feel free to make adjustments to it if you wish.