It was once the case that those embarking on a study of the Greek New Testament would have already had a thorough grounding in Classics. Such is no longer the case, but the earlier situation is reflected in two excellent — though quite different — works on the language of the Septuagint.
A brief study of the tranlsation Greek of the Septuagint appeared in a Selections from the Septuagint: According to the Text of Swete by F.C. Conybeare and St. George Stock (Boston: Ginn & Co., 1905), written in the UK, but published in the USA! The grammar section alone was reprinted by Zondervan in 1980 (and subsequently), but the general introduction and the numerous selections (all heavily annotated) is still worth consulting. The whole volume is available in a clear scan by Google Books, but the download link is no help — get it here instead [direct link to PDF].
A much more substantial work appeared a short time later, written by the indefatiguable translator of Josephus, Henry St. John Thackeray. He only produced the first of a projected multi-volume project which covers orthography and accidence. Still, it makes available Thackeray’s rich experience of a wide range of Greek texts brought to bear on the Septuagint. Archive.org has two good scans, although one includes bookmarking of the contents which makes it the more usable (or, check out the other one).
Study of the Septuagint’s Greek has not stood still! For a brief but fairly recent overview, see Jenny Dines, The Septuagint (London: T & T Clark, 2004), pp. 110-117.