I would like to do the following:
I have an existing .jpg picture from a Webcam.
Now I would like to overlay this Picture with the following Widget:
<div style="width:250px"><a href="http://wetter.webmart.de/782303" target="_blank" style="text-decoration:none">Wetter in Eglisau</a><br /><script src="http://wetter.webmart.de/e/782303"></script></div>
Then the picture should be saved under a new name in the same folder.
Can I do this with a PHP Script?
I have searched tutorials, but the are all for only text or Image over Image etc.
Unfortunately, php can't help you with this. If you need to produce just one single picture, then image editor will do.
But I guess, you need to automate this process with fresh webcam images and weather updates. I would recommend to create a web page with two image as @MilanG suggested using css tricks. After that you can use some headless browser to capture the screen, e.g. PhantomJS http://phantomjs.org/ will do. Again, as I wrote in the beginning, php cannot help you with this task.