in the WoW client folder is a file called realmlist.wtf. That file can be changed to connect to real or private servers. Create a backup of this file. Set up a private server using any of the guides. I recommend looking in the forums for a tutorial. You will need to set up a private server that supports the latest patch version (so you can use the same client).
Just switch realmlist.wtf with one to connect to your private server, then switch back to it to connect to real WoW. Back up the original version of this file so that you can switch back. It's a pretty small and simple file. Open it in notepad, and you will see there are only two lines in it. These lines tell the client what server to connect to for game play and for patches. Tutorials will tell you what you should put in this file to connect to a private server. Good luck.