1.Open Bitcore
2.Click file
3.Sign message.
follow screenshot below.
When you click 'sign message' it will ask for your passphrase.
For the October 30th BTC Snapshot. I would have to use my BTC private key to claim the requivalent BTX?
The private key are only yours to keep. You do not give out your private key. If you're talking about your BTC for redeeming your Bitcore, you'd have to submit a signed BTC address to the the bitcore website. Your BTC wallet should have a signing feature/capabilities to be able to do this step.
can I claim them direct to an exchange address? (The april snapshot)
If you're talking about claiming the bitcores to an exchange address, as far as I know, that won't work, It has to be claimed by using your bitcore wallet address that you'll have to download to your desktop computer.