1580s, in the rhetorical sense ("a chain of reasoning in graduating steps from weaker to stronger"), from Late Latin climax (genitive climacis), from Greek klimax "propositions rising in effectiveness," literally "ladder," from suffixed form of PIE root *klei- "to lean."
Originally in rhetoric an arrangement of successive clauses so that the last important word of one is repeated as the first important word of the next, as in Romans v.3-5: "... but we glory in tribulations also: knowing that tribulation worketh patience; and patience, experience; and experience, hope: And hope maketh not ashamed ...." Compare anadiplosis. From the rhetorical meaning, the word evolved through "series of steps by which a goal is achieved," to "escalating steps," to (1789) "high point of intensity or development," a usage credited by the OED to "popular ignorance."
The meaning "sexual orgasm" is recorded by 1880 (also in terms such as climax of orgasm), and is said to have been promoted from c. 1900 by birth-control pioneer Marie Stopes (1880-1958) and others as a more accessible word than orgasm (n.).