I set up my Linux system to type Greek polytonic characters. It’s Ubuntu 16.04, Unity desktop environment, the default. My locale is set to “en_US.UTF-8”; my physical keyboard is an ordinary American keyboard. Here is what I did, expressed as instructions.
Find “System Settings”, and click on “Text Entry”.
Go to “Input sources to use”, and click “+”
Find “Greek (polytonic)” and select it.
Choose keys for switching input sources. My system has two input sources, “English (US)” and “Greek (polytonic)”. I use Control+/ to change input sources.
Fiddle with the other choices to decide what you want.
Needless to say, the page shown has nothing to do with setting up a Greek keyboard in Linux! It’s probably from a tax FAQ on alternate ways of providing tax documentation on sales from gas stations of liquid fuels to businesses (to obtain a rebate)
FWIW, since I posted this, I have decided that I prefer to type polytonic Greek in Emacs, using the Emacs input method greek-ibycus4. You can have lots of files open in Emacs, but when you set an input-method, it is active only in the buffer(file) you are using when you set the input method. All the other buffers (files) remain as they were as far as keyboard input is concerned.
I don’t type much Greek. I do a lot of Greek handwriting, for example copying sentences into a notebook for parsing and for grammar study.