the country which gave a visum is responsible for the asylum procedure during the validity of this visum and 6 months after its expiring date. That means, your friend must stay illegal somewhere this whole period, which is probably difficult. If the police catches her, she will probably get a criminal procedure because of illegal staying in Germany (illegaler Aufenthalt). If she leaves Europe (the space of Dublin states), comes back being able to give a proof that she was away, and the visum is expired, then Greece wouldn't be responsible any more (in this case 6 months after expiring don't matter). The problem is that she must find a possibility to come to Europe once more and then she must come directly to Germany if she wants Germany to be a responsible country from the very beginning.
If she starts her asylum procedure in Germany now, she will, with most probability, get a rejection "Asylum application inadmissible" from BAMF, with the threat of deportation to Greece. She can claim against it in the administrative court. It is difficult to evaluate her chances. If she has serious health problems and can prove it, her chance that the court permits her to stay in Germany is perhaps not bad. The courts in Germany decide very differently, but there are many positive decisions about Greece, especially for vulnerable persons.