Conducting individual research under supervision of a "Doktorvater" or "Doktormutter" is the typical German way of obtaining a PhD.
The writing of a doctoral thesis or dissertation is the core element of the doctorate and usually takes 3 - 5 years. Students agree their topic with a supervising professor (Doktorvater or Doktormutter) and then write a thesis or dissertation as a piece of independent research work.
The main steps involved in this area: Finding a supervisor from the website of the educational institution or from publications by the professor that you came across in the relevant field that matched your interest. Contact your future supervisor by E-Mail or formal letter and provide him/her with your CV/Resume elaborating on your background, relevant work-experience and skills in addition to your planned research project.
If you follow this way of gaining your PhD you have to pay attention to financing your studies. You'll normally have to fund your individual research yourself/with a grant. There is also a possibility of working at a department of the University as a research assistant ("wissenschaftlicher Mitarbeiter") and writing your thesis on your work. For this option it is an advantage to speak German. If you are interested in such a position, you can check whether there are any open positions in the respective departments.