One of the primary problems with considering deconstruction a "method" is that it effaces the complexity and nuance of the actual way by which deconstruction occurs, and reduces it into a mechanistic operation that a reader can perform on a text. Instead, deconstruction is something that happens within the text, and an outside reader then interprets the results of said process. If you were to perform a "deconstructionist reading of The Great Gatsby", for instance, you would first look to canonical readings of the text, and see what sorts of definitional frameworks, and thus, binary oppositions, such readings of the text depend on. A "deconstructionist reading" would require considering what would happen if these binary oppositions either did not exist, or were flipped with the marginalized term coming to the center, with the non-privileged term gaining power over the traditionally powered term. The resulting multiplicity in meaning that is produced within the text, and the reduction of a violent power hierarchy to immanence is deconstruction. To clarify, deconstruction is NOT what systemic reconsiderations the reader makes to ultimately arrive at multiplicity in meaning. Deconstruction is what the text does after the reader has made this reconsideration.
To provide an example from Derrida's "Structure, Sign, and Play in the Discourse of the Human Sciences", Derrida talks about the anthropologist Claude Lévi-Strauss's works on the incest taboo. Lévi-Strauss, an anthropologist, is focused heavily on the development and arising of human culture in contrast to its natural surroundings, and thus many traditional readings of his work rely on the binary opposition of "culture/nature" with culture taking privilege over nature. Yet, Derrida identifies a moment in the text that contradicts the assumptions made by these traditional readings, and thus shows how the text itself of Lévi-Strauss's work resists all of these traditional readings. This moment is where Lévi-Strauss describes the taboo on incest as a cultural construct, thus belonging to the side "culture" of the binarism, but simultaneously explains that the incest taboo is a universal phenomenon, and is thus natural - belonging to the "natural" side of the binarism as well. Lévi-Strauss's incest taboo is an example of what Derrida would call a lost middle, since it does not neatly fit onto one side of the binarism, and calls into question the definitions and frameworks that assert "culture is not nature" and "nature is not culture". So, what Derrida did - sit down at his desk, open up his copy of The Elementary Structures of Kinship by Lévi-Strauss, and underline some passages in the text that show the incest taboo being a lost middle - is not deconstruction. Instead, the little foible in the text and its calling into question traditional readings of Lévi-Strauss is deconstruction.
And, as Derrida did, I will also offer a disclaimer on what I have written above. Derrida has long since argued that attempting to simply state, "deconstruction is [X]" effaces its complexity and replaces the idea of deconstruction with some verbiage and weak conceptual understanding that can sate a desire to learn what deconstruction is. As Paul de Man argued, to understand a concept is effectively to feign understanding and beguile oneself into complacency with a reduced, watered-down explanation. It is for this reason Derrida himself is very careful about offering examples with which the theory itself can be supplanted with. But this observation also helps to further explain why Derrida resented calling deconstruction a method - for replacing deconstruction with a formulaic process is a surefire way to do exactly what he feared: reducing, oversimplifying, effacing nuance.