After some searching and testing how to debug a windows service, I found that for me the following solution is working very well.
First register your service as a service on your machine. This can be done with a few simple steps. How to do this is described in this article: http://kannekens.nl/registering-installing-a-windows-service/
Next add a line / lines Debugger.Break(); in your code where you want the debugger to start. Now compile in debug mode clicking in Visual Studio menu: Build, Build Solution while the Configuration Manager is set to Debug.
After the application compiled successfully we can stop and start the service to ensure these modifications are run:
I used the tips from https://docs.microsoft.com/en-us/dotnet/framework/windows-services/how-to-debug-windows-service-applications and https://stackoverflow.com/questions/104235/how-can-i-use-debugbreak-in-c.
Right click the Visual Studio application and select more…, Run as administrator
Click Debug, Attach to Process in the Visual Studio menu.
This will open a dialog where you need to select the right process to debug. Check the Show processes from all users checkbox. Find your Service process and click Attach.