There was a recent question on intuitions about sheaf cohomology, and I answered in part by suggesting the "genetic" approach (how did cohomology in general arise?). For historical material specific to sheaf cohomology, what Houzel writes in the Kashiwara-Schapira book Sheaves on Manifolds for sheaf theory 1945-1958 should be adequate.
The question really is about the earlier period 1935-1938. According to nLab, cohomology with local coefficients was proposed by Reidemeister in 1938 (http://ncatlab.org/nlab/show/history+of+cohomology+with+local+coefficients). The other bookend comes from Massey's article in History of Topology edited by Ioan James, suggesting that from 1895 and the inception of homology, it took four decades for "dual homology groups" to get onto the serious agenda of topologists. It happens that 1935 was also the date of a big international topology conference in Stalin's Moscow, organised by Alexandrov. This might be taken as the moment at which cohomology was "up in the air".
Now de Rham's theorem is definitely somewhat earlier. Duality on manifolds is quite a bit earlier in a homology formulation.
It is apparently the case that At the Moscow conference of 1935 both Kolmogorov and Alexander announced the definition of cohomology, which they had discovered independently of one another. This is from http://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf at p. 11, which then mentions the roles of Čech and Whitney in the next couple of years. This is fine as a narrative, as far as it goes. I have a few questions, though:
1) Is the axiomatic idea of cocycle as late as Eilenberg in the early 1940s?
2) What was the role of obstruction theory, which produces explicit cocycles?
Further, Weil has his own story. Present at the Moscow conference and in the USSR for a month or so after, his interest in cohomology was directed towards the integration of de Rham's approach into the theory. He comments in the notes to his works that he pretty much rebuffed Eilenberg's ideas. Bourbaki was going to write on "combinatorial topology" but the idea stalled (I suppose this is related). So I'd also like to understand better the following:
3) Should we be accepting the topologists' history of cohomology, if it means restricting attention to the "algebraic" theory, or should there be more differential topology as well as sheaf theory in the picture?
As said, restriction to a short period looks like a good idea to get some better grip on this chunk of history.