So your media are on the external drive and your library files are on the internal? iTunes is a bunch of files, not just your media files.
What are the iTunes library files? - http://support.apple.com/kb/HT1660
More on iTunes library files and what they do - http://en.wikipedia.org/wiki/ITunes#Media_management
What are all those iTunes files? - http://www.macworld.com/article/139974/2009/04/itunes_files.html
iTunes 9 [and later]: Understanding iTunes Media Organization - http://support.apple.com/kb/ht3847
If your library files are on your internal drive you need to copy the iTunes folder from Home > Music to the same location on the new computer (make sure you don't mind replacing the one already there), then start iTunes. It may want to update your main library file.
If instead of what you say, the whole iTunes library is on the external drive:
iTunes: How to move [or copy] your music [library] to a new computer [or another drive] - http://support.apple.com/kb/HT4527
Quick answer if you use iTunes' default preferences settings: Copy the entire iTunes folder (and in doing so all its subfolders and files) intact to the other drive. Open iTunes and immediately hold down the Option (alt) key (shift on Windows), then guide it to the new location of the library.itl file in the moved iTunes folder.