Is the external drive permanently connected?
Storing the iTunes Library on an external drive can be done but can be hazardous also. You need to be extremely careful you do not open iTunes if the external drive is not connected or active.

You can go to iTunes preferences settings via Edit menu. Click on Advanced tab. Click Change button to change location. Copy files to iTunes Library when adding to iTunes option should also be checked.
I can't remember if iTunes manually copies the library to the external drive or you need to do this manually. iTunes will keep a library data file that keeps track of location either within your user preferences or within the iTunes music library folder on the computer. This file must remain there.

If you do change location to an external drive, be careful at each step of setting it up & about the warnings I gave above. It's been several years since I last did this so I cannot recall the specific details. Nowadays on my new computer I use a partition specifically for iTunes Library (ie: using 1 TB HDD's split into specific purpose partitions. My iTunes partition is 450 GB which would be overkill for most people. I've only used half of this so far.)