Does it mean in order to avoid the replay attack, we need to kill one chain so that there is only one chain to do the business.
No, absolutely not. From the moment that you do a transaction that contains even just some dust of newly mined coins on one chain, that transaction is not valid on the other chain and you are from that point on, with that address, only on one chain.
With ETC/ETH, you can also use a contract (there's a blog post by Vitalik explaining this).
But in general, on any block chain, things go like this:
Pre-chain: you have UTXO ("addresses") of which you possess the secret key that can sign transactions.
If you sign a transaction using ONLY these UTXO with that secret key, you are making a new transaction to new UTXO (the "recipient address").
On both chains, your signature will be valid (some people call this automatic feat a "replay attack", I wonder why), and your transaction will be included if you do not pay attention. If that is the case, then on both chains, the former UTXO is not unspend any more, and the new address is now an UTXO. Warning: any signature on any chain of THIS POST-fork transaction is ALSO VALID on both chains and will ALSO be included on both chains !
So what can you do (in general, on any forked chain, not just ETH/ETC):
NEWLY MINED coins that appear in blocks after the split are of course only valid UTXO on one single chain: they are not considered valid on the other chain, because these are coinbase transactions in blocks that do not appear on the other chain. *if you can get hold of some of these, even a minor amount* and you now sign a transaction with a COMBINED INPUT containing some of this mined coin dust and your former coins, then THIS TRANSACTION and its signature will of course NOT be valid on the other chain.
So this transaction will NOT turn the former UTXO into a spend output on the other chain, because no valid transaction uses it.
And from that moment on, the resulting new UTXO on the first chain will also only appear and be valid on one chain. From that point on, whatever you do with those UTXO, you will never have to worry again about them initiating a transaction on the other chain because these UTXO don't exist on the other chain.
So in as much as coins will get more and more mixed with newly mined coins on each branch, this phenomenon of valid transactions on both chains will disappear.