Deconstruction is frequently described as a method of reading, but Derrida himself resisted this description because "method" implies a set of procedures you can learn and apply mechanically. Deconstruction is not algorithmic. It requires genuine attentiveness to the specific movements of the specific text being read, following the text's own logic to the point where it outstrips or contradicts itself.
That said, there is a recognizable movement in Derrida's practice that can be described. It has roughly two stages:
First, identify the hierarchy: almost every philosophical text, every serious piece of reasoning, relies on a conceptual hierarchy in which one term is treated as primary, positive, and foundational, while its apparent opposite is treated as secondary, derivative, and possibly threatening. Speech/writing. Presence/absence. Literal/figurative. Nature/culture. Inside/outside. Serious/playful. Original/copy. The hierarchy is not usually announced, it is assumed, built into the structure of the argument.
Second, follow the text's own logic to the point where the hierarchy inverts or implodes: show that the text cannot do what it needs to do without relying on what it demotes. That the supposedly secondary term is in fact constitutive. That the hierarchy is unstable not because you are importing a critique from outside but because the text itself, followed carefully enough, cannot sustain it.
A beautiful and accessible example is Derrida's reading of Plato's Phaedrus in the essay "Plato's Pharmacy" (in Dissemination, 1972). Plato's Socrates argues that writing is dangerous, a pharmakon, because it is a substitute for genuine memory and living dialogue. The written text cannot answer questions, cannot tailor its message to its audience, cannot defend itself from misreading. It is a dangerous supplement to the living speech of genuine philosophy.
Derrida notices that the Greek word pharmakon means simultaneously poison and remedy, it is an irreducibly ambiguous word that the standard translations have always rendered as one or the other, depending on context, suppressing the ambiguity. Derrida reads the dialogue as organized by this undecidable pharmakon: writing is a poison that threatens the purity of living philosophical dialogue, but it is also a remedy, Plato has written the dialogues in order to transmit Socratic philosophy. Means that the philosophical tradition Socrates was part of survives only because of the writing Socrates distrusted. The dialogue that argues against writing is itself a piece of writing. The condemnation of the supplement is itself supplementary.
This is not a cheap gotcha. It is a revelation about the structure of Plato's situation: the philosophical tradition requires both the ideal of pure living dialogue and the supplement of writing that keeps it alive, and these two requirements are in irreducible tension. The deconstruction does not destroy Plato's argument. It shows what Plato's argument requires but cannot acknowledge.
A second example is Derrida's reading of Rousseau in Of Grammatology (1967). Rousseau's account of language treats speech as natural and prior, writing as artificial and supplementary, a corruption of the pure original voice. But as Derrida shows, Rousseau's own descriptions of natural speech turn out to be organized by the very structures he attributes only to writing: citability, repeatability, distance from original context. The natural state that writing supposedly corrupts was never available in the pure form Rousseau imagined. The supplement was always already there, built into the structure of language from the beginning. The origin was always already contaminated.