I installed the tool as mentioned but it is not doing anything at all.
What exactly did you install, the ZIP of the stand-alone version or the OXT for LibreOffice/OpenOffice?
I also installed a runtime environment before that
Have you checked the common problem list at https://languagetool.org/issues/?
I repeated everything again (deinstalled and reinstalled). I had to restart the programm 3 times now it works. I tried the example sentances but it made no reaction after closing for the first 2 times.
I did not do anything different than restarting programm more than once. Is it normal to restart several times?
One restart should be enough (unless you have some kind of quick starter - I'm not sure if those still exist, in those cases you also need to quit the quick starter).
Well it works after closing the programm 3 times, maybe it was a hidden window in another mode. My Windows 8 System is very new and I am not used to switching between start and desktop modus.
Thanks for the nswers!