What is the best way to validate qualitative research which is based on case studies? How can the researcher achieve internal and external validity?

Validity is questioned within qualitative research however by ensuring you use a stringent and sytematic analytical analysis process that is clearly explained in the methodology section you should overcome this. Transparency and reflexive practice of your process will help.  You could assess how the previous research was conducted and how they arrived at their findings then confirm or refute their approach. The findings are also dependent on the aims and objectives of the research so they will also have to be taken into account within the process.