How can I kill Zelkarons without the Silver chimes?

You don't actually need to have the ring equiped at this stage, so you can switch to the chimes instead. You equip or unequip it from the inventory screen the same way you would equip the underwater breathing necklace or use the crystal or the shell. Equiped items have the blue background in their slot.

Killing the Zelkarons is a bit tougher matter, but it's just a question of hacking and slashing them for a bit, or gaining a level or two (or more) and returning.

Only AFTER the ring is charged you need to have the crystal necklace equiped not to die while putting on the ring.

You can't: you have to use the silver chimes. My understanding is I need to kill one while holding the Blue Necklace instead is wrong:

There actually are a couple of seconds between the "Evil Ring has charged" message and the 9,999 damage that kills you. If you open the menu immediately after the message appears and equip the Blue Necklace, you will not take damage.