Best Universities for Astrophysics
Many German universities have very attractive PhD astrophysics programmes. Currently there are a total of and most of them offer the possibility to pursue a doctorate.
Among these various choices available in Germany, I would however single out the international programmes with a structured curriculum taught entirely in English, in particular the International Max Planck Research Schools. They combine scientific excellence with an infrastructure specifically designed to provide the best support for students coming from abroad to study and do research in Germany. A search on the pages of the turns up the Bonn-Cologne Graduate School of Physics and Astronomy, Astrophysics in Munich, Solar System Science in Göttingen, Astronomy and Cosmic Physics in Heidelberg, and Astrophysics in Potsdam, all of which I can indeed highly recommend for their quality of training and education.
Although the names sound alike, these programmes are centred around different research areas (and have reputations for different specialisations), so I would always suggest to closely look into which advisor/research project combinations best match your own interests and skills before applying. In Bonn, for example, the interests of many faculty members are centred around radio astronomy, whereas you will find that the solar physics and solar system focus in Göttingen extends to extrasolar planetary systems and their host stars, amongst other things. (Disclaimer: The Solar System School in Göttingen happens to be where I am working - on asteroseismology and exoplanets -, although my host institution is probably best known these days for its heavy involvement in the Rosetta mission.)
Since the structure of graduate eduction in the USA differs somewhat from that in Germany, one consideration in choosing a location may also be a candidate's current formal level of education - in the North American educational system, graduate school entrance usually requires a Bachelor's degree, whereas most German PhD programmes actually require a completed Master's degree, unless the programme has implemented a fast track option.