How can I reclaim my Bitcoin Cash from my Greenaddress wallet? I'm new to cryptocurrency in general, and I'm still learning.  

Let's say someone moves BTC to a multi-signature wallet with a trusted provider, i.e. Greenaddress, before the split from an exchange. 

This now does not provide him with Bitcoin Cash too. 

Can someone please explain the process of obtaining the Bitcoin Cash that goes with his wallet's Bitcoin?

It seems that one can move to a wallet that does support Bitcoin Cash. 

But I'm not clear on the concept of the transfer of private keys/addresses from a wallet that does not support Bitcoin Cash to a wallet that supports Bitcoin Cash.

Is this possible, and how does it work? 

Would this transfer happen by simply converting Bitcoin to BCC, or would the BTC holder keep the BTC balance and gain a BCC balance in addition?
Unfortunately, if your wallet/exchange does not support BCC, then you have probably forfeited it i.e. left it to their disposal. The only way to find out would be to message their support and see if it is supported. If it is, you will automatically have a balance of both. There are a few exchanges supporting BCC trade including bitrex.com (or have a look at coinmarketcap.com &rarr;     Bitcoin Cash &rarr;     Markets)