I don't know about Dogecoin Core. but I can tell you how you would do it in Bitcoin Core.
First you need to unlock the wallet with walletpassphrase command for n amount of seconds. In this example you will unlock it for 10 minutes (n=600).
bitcoin-cli walletpassphrase "yourpassword123" 600
Then you can use dumpprivkey to get a private key for your address.
bitcoin-cli dumpprivkey "1YourBitcoinAddress"