Short update: The museum in Berne pointed me to a book with a small family tree, but it only has the years. Next step is to get the civil records of Berne, and then move on to the other cities.
It's quite a complicated family, as both father and wife of Paul Klee were from Germany, but of course not the same areas. Pauls mothers parents were from Basle (but she herself was born in France), there I will continue my research.
There already is a photograph of Paul Klees grave linked in his profile, so I will skip that.
If someone has time and interest, please feel free to correct Paul Klees biography. My English is not so good and it takes me ages to write a long text.