Get W. Sidney Allen's slim text _Vox Graeca_. It goes over not only how Ancient Greek was pronounced but also how we know. The evidence includes explicit notes from grammarians and other writers, graffiti, borrowings, puns, and later forms of Greek. Allen also has another book _Vox Latina_ which does the same thing for Latin.

A good question. The best, i.e. least likely to be unreliable, sources are: epistles as preserved by papyri; conversations as preserved in ancient drama (especially comedy); phrase-books that survive from ancient school curricula; funerary inscriptions that use everyday language; graffiti.