I know Edith Hamilton best as the English translator of the most common edition of Greek and Roman myths--the famous edition of her Mythology is typically assigned in high schools. When I saw her book The Greek Way on the fundraising book sale table at work recently, I had to have it. It was published in 1930 and 1942 (not sure how that works--maybe certain parts appeared first in magazines?) and is a collection of critical essays on Hellenic-age Greek literature (i.e. the 5th and 4th centuries B.C.).
She moves more or less chronologically through ten of the most important Greek authors, starting with the poet Pindar and ending with the dramatist Euripedes, with a few essays on special topics like the religion of the Greeks or the Greek way of writing thrown in. She takes the characteristic she finds most important with each of these ancient authors and highlights it, typically by comparing him to a more modern author--the raw, heroic tragedy of Aeschylus with Shakespeare, for example, or the "exuberant, effervescent" comedy of Aristophanes with Victorian light opera librettist W.S. Gilbert (of Gilbert and Sullivan)(!).
My favorite essay was on Xenophon, a Greek writer I was only slightly familiar with beforehand, whom Hamilton shows as the great portrayer of everyday Greek life, particularly for the leisured class in Athens. Xenophon is practically our only source outside of Plato to give an account of Socrates, but is also known for his autobiographical account of the retreat of 10,000 Greek mercenaries from Persia back to Greece after the general who had hired them was cut down in battle. The enemy didn't even bother mopping them up, assuming 10,000 leaderless foreign soldiers would simply perish, but that was not the case. Greek individualism and common sense carried them through a two-year journey across deserts and mountains, until they finally reached the Mediterranean Sea with most of their company intact. It is quite a stirring story.
I am not quite sure I could recommend this book generally. I think it would probably be most interesting to those who have somewhat more than a passing knowledge of ancient Greek literature and history. I found it quite fascinating myself, and my guess is if it sounds like a book that would interest you, you're probably right.