NT Wright on Paul’s use of “flesh”:
But what do “fleshly”…mean? ….[“Flesh”] is so problematic that it would be nice (as I have tried to do with some other technical language) to avoid it altogether, but I have found that doing so produces even worse tangles. Better to learn, once and for all, that when Paul uses the word “flesh” and other similar words he does not intend us simply to think of the “physical” world, in our normal sense, as opposed to the “non-physical.” He has other language for that. The word we translate, here and elsewhere, as “flesh” refers to people or things who share the corruptibility and mortality of the world, and, often enough and certainly here, the rebellion of the world. “Flesh” is a negative term. For Paul as a Jew the created order, the physical world, was good in itself. Only its wrong use, and its corruption and defacing, are bad. “Flesh” highlights that wrong use, that corruption and decay.
Reposted from New Leaven. pp. 140-41, Romans 1-8, Paul for Everyone.
God is Open 
Paul uses “flesh” a lot and he uses is flexibly. At least in one case I can think of he uses it to mean human nature, though he generally avoids “nature” in the theological sense. Furthermore Paul draws a lot of dichotomies by way of contrast, flesh vs spirit, law vs grace, grace vs works. His anthropology doesn’t seem to be rigorous but is sketched out by way of these comparisons: natural man vs spiritual man, carnal mind vs spiritual mind, even mind vs flesh, mind vs body. The author of Hebrews goes one step further separating “soul from spirit” explicitly in Hebrews 4:12, something Paul does only implicitly.