Optional: Nameless King
Maximum souls: 594,864
The most difficult aspect of the fight is the camera issue in the 1st phase against the Storm Drake. Use crystal sorcery to quickly dispatch of the 1st phase. Equip the Court Sorcerer's Staff+10 in the right hand and the Scholar's Candlestick in the left hand. To buff the staff, you must 2-hand it and press L2. You can easily buff the staff by equipping it in the left hand and pressing L2; no need for 2-handing. Equip these to boost your damage output: Crown of Dusk, Magic Clutch Ring, Young Dragon Ring, Bellowing Dragoncrest Ring. The 4th ring can be any of your choosing, but the Chloranthy Ring+3 is recommended as it would also help you in the 2nd phase. Use Yorshka's Chime+10 to cast Sacred Oath and Deep Protection, buff your staff, and cast Homing Crystal Soulmass. Lock onto the head of the Storm Drake and use Crystal Soul Spear as your primary offensive weapon.
In the 2nd phase, use the Blood Bandit's Knife+10 as your primary offensive weapon to bleed the Nameless King. At the start of the 2nd phase, quickly swap to rings that boost your defense such as Ring of Steel Protection+3 and Thunder Stoneplate Ring+2. Equip the Blessed Caestus+10 in the left hand and the Sun Princess Ring to help with your HP regeneration. Consume a Yellow Bug Pellet to boost your lightning defense.
The above strategy requires you to know how to quickly swap between different weapons and rings. Before starting the fight, you might as well equip these to save some time in the 2nd phase: Blood Bandit's Knife+10, Court Sorcerer's Staff+10, Scholar's Candlestick, Blessed Caestus+10. See the following video for the sorcery and bleed strategy.
Nameless King NG+32 crystal sorcery and bleed, Dark Souls III
One strategy to minimize the swapping of weapons and rings is to use Great Lightning Spear against the Storm Drake and a bleed weapon against Nameless King.
Nameless King NG+34 Great Lightning Spear and bleed, Dark Souls III
Another strategy to minimize the swapping of weapons and rings is to use the Lothric Knight Greatsword+10 to fight the Storm Drake. Prior to entering the boss arena, cast Sacred oath and Deep Protection, and use Yorshka's Chime+10 to buff the sword with Lightning Blade. Equip the Ring of Steel Protection+3 and Thunder Stoneplate Ring+2 to boost your defense. Also consume a Yellow Bug Pellet to boost your lightning defense. Equip the Blessed Caestus+10 in the left hand and the Sun Princess Ring to help with your HP regeneration. R1 the head of the Storm Drake. The 2nd phase is the same as per above. The video below demonstrates this strategy.
Nameless King NG+32 Lothric Knight Greatsword & bleed, Dark Souls III