In ancient Athens, calling someone a hypocrite was not an insult. It was a job description. A hypokritēs was an actor.
The word grew out of the verb hypokrinesthai. Etymonline traces its sense through a chain of steps. It began as "separate gradually," built from hypo-, "under," and krinein, "to sift, decide." It then came to mean "answer," then "answer a fellow actor on stage," and finally "play a part." An actor, in other words, was first of all an answerer: someone who spoke back in dialogue. Merriam-Webster gives the same range for the verb, from "to reply" to "play a part on the stage" to "feign."
The trouble was that acting and pretending are close neighbors. Wikipedia notes that hypokritēs was a technical term for stage actors, a trade thought unfit for public figures. In the 4th century BC, the orator Demosthenes used it against his rival Aeschines, who had been a successful actor. The jibe was that a man so good at playing characters could not be trusted in politics. The noun hypokrisis, "acting on the stage," was already shading into "pretense."
The New Testament finished the shift. In the Gospel of Matthew, Jesus condemns the scribes and Pharisees as hypocrites. These were people performing piety in public while their inner lives did not match. That religious sense is the one Latin picked up as hypocrita.
English borrowed the word from Old French ypocrite around 1200. Its first English meaning was specific: "a false pretender to virtue or religion." The Greek "h" had dropped out along the way, and English restored it in the 1500s.
Over time the religious edge wore down. Merriam-Webster's second sense is broader: anyone whose behavior contradicts their stated beliefs. A hypocrite no longer has to be pious, only inconsistent.
The theatrical origin still fits surprisingly well. A hypocrite is someone playing a role, and the insult lands because the audience has noticed the costume.