Navigate into the Eleos wallet and click on the "Addresses" tab. Click on the "Save private key" button and you will be prompted to select the directory that you wish to save the private key file to. It is important to give a name to the file and have the file end in a .txt format in order for the file to be able to open and display the keys properly, as the file comes in a blank file extension by default. Make sure you save your private key file somewhere safe and secure, such as an offline USB where nobody can access it. Also, do not share the keys to anybody.
Once you've exported your private keys file, open up the text file and you should see an entire dump of your zclassic eleos wallet. This includes all the addresses and its corresponding private key you've generated in the wallet up until the point in time you exported the private key file. Note that if you generated an address and used it AFTER you exported the private key file, you will need to export it once again, as the new address and its private key has not been saved.
If you need further assistance, please refer to this extensive guide on the Eleos wallet for troubleshooting and extracting the private key from a wallet file.