It is notoriously difficult, when discussing periods in which early death, and sudden death caused by infection or disease, could carry off even the strongest and healthiest with very little warning, to determine when a death was caused by poison.

That said, there was an extensive literature on poisoning. Doctors would be looking for specific symptoms, among which were excessively violent vomiting and much drier stools, and, on autopsy, blackening or withering of the bowels. I touched on some of these issues in considering poisoning scares in the early modern period in an essay here, and although 1500 or so years had passed since the period you are interested in, many of the issues and the diagnoses actually remained the same.

I'd also suggest checking the essays in Philip Wexler [ed.], Toxicology in Antiquity for more detailed analysis of how antique diagnostics worked in cases of poison.