There's a few things you're missing here in order to have the computer run completely off of the external HDD. While it is possible to do (boot from USB device etc..) simply copying all of the contents to the drive will not accomplish what you are looking to do. In order for the drive to be bootable, it needs a working MBR (master boot record) cloned from the original/internal drive and the easiest way to do that is with imaging software. This specialized type of software clones drive structures so they can be applied to other drives and/or partitions.
Using the image software like True Image is very easy and is by far the most effective way of cloning disks quickly and efficiently. Once the external drive, which is actually the same exact thing as an internal drive (it's just in it's own housing), has been written with the partition data and MBR from the original drive, you should be able to boot from it easily. It sounds a lot more complicated than it really is and can be done in a matter of a few hours depending on the size of the data partition and how fast the hardware is.