Which language did Cleopatra's dynasty speak at court?

Greek. The Ptolemies were Macedonian, descended from one of Alexander's generals, and ruled Egypt for nearly three hundred years while conducting government in Greek - Alexandria was a Greek city with a Greek administration sitting on top of an Egyptian country.

Cleopatra VII was reportedly the first of the line to bother learning Egyptian, which is remarkable after nine generations. Plutarch says she spoke many languages and rarely needed an interpreter, listing Ethiopian, Hebrew, Arabic, Syriac, Median and Parthian among them - and the political point of learning Egyptian was that it let her present herself to her own subjects as a pharaoh rather than as an occupier, which she did deliberately, appearing as Isis.

The linguistic split is also why the Rosetta Stone exists. It carries the same decree in Greek, in Egyptian demotic and in hieroglyphs, because an administration operating in Greek had to publish to a population that did not read it - and that redundancy, produced by a governing class speaking a different language from the governed, is what let Champollion read hieroglyphs two thousand years later.