Essentially, you need to move your BTC to a new address/sweep it, load your previous private key(s) to the fork coins wallet, and then that's it.
There's a complication if you're claiming Bitcoin SV: it doesn't have replay protection, which means that if you split your Bitcoin Cash you have to take precautions to prevent the Bitcoin SV from moving too.
As long as you're moving between your own wallets it can be recovered, but if you decide to send your Bitcoin Cash to an exchange, the Bitcoin SV will move with it. Coins sent to an exchange on the wrong chain are usually not recovered.
An easy fix is to get a small amount of BCH or BSV that only exists on that chain, and include that input in the transaction with your own coins. If you do this, the transaction can only exist on one chain.