Simply connect the CCC EHD and drag the files you want to the new computer. However if it were my computer I'd simply use Setup Assistant and only migrate my data files. However since you already have the computer and I am guessing have set it up then locate the libraries you want and copy and paste them to the new computer. Apple has advice letters on advising how to move iTunes, iPhoto and iMovies libraries to an EHD. I'd Google those, then you can tell where the libraries are located and from that it's a simple drag and drop process. Your Mail and other files you will need to locate using Finder and then move those libraries manually.
IMHO you are making life more difficult, I'd recommend simply using Migration Assistant.