The most obvious reason for getting a 100061 "Connection Refused" error is if there isn't a server listening at the remote end.
And as you say that you're using the Client example for the Arduino then that's true - there won't be anything listening on the Arduino.
TCP isn't quite like a serial connection where you just open the serial port at each end and can send and receive data. It's a client-server model where one end initiates the connection (the client) and the other end must already be listening for that incoming connection (the server).
It sounds like you're using something like telnet on your PC, which is effectively trying to connect two clients together, and won't work.
The easiest solution is probably to use the ChatServer example on your Arduino. Then the Arduino will be acting as a server, listening on port 23, and you'll be able to telnet to it from your PC.
Anything you type into telnet will then be echoed back to you by the Arduino. It might not look like anything much is happening at first, because you'll just see what you're typing, but because of how telnet works the text will have been sent to the Arduino and back if it's appearing :-)