If you plan to kill more than one of the kings, you will need all three attack styles as they are all immune to two of the three, with Supreme being immune to Magic and Ranged. However, even if you're planning on killing just one king, you should still bring all three styles in case one or both of the other kings spots you and needs to be killed.
It is possible to repeatedly kill just one of the kings without the other two attacking you as each king tends to stick to one area of the island and won't spot you unless you wander into their area or stand in the middle. To kill Supreme, you should wait until Prime and Rex have walked away from the middle and lure him to the ladder to prevent being spotted by the other two.
As Supreme can be damaged with melee, it is possible to use the Balmung to deal extra damage.
Wearing Fremmenik sea boots 4 while fighting the kings will increase your damage by 10% (5% for abilities) and note some of their drops.
For more information on fighting Dagannoth Supreme, see the Dagannoth Kings Guide.
The chance of receiving better drops is increased by the Ring of wealth, Luck potion, Collector's insignia, Leprechaun hat or Luck of the dwarves.