The first in-canon mention of Vader's true name (within the original unimproved trilogy) is in RotJ, during the scene where Luke speaks with the Force-ghost of Kenobi.
LUKE: You told me Vader betrayed and murdered my father.
BEN: You father was seduced by the dark side of
the Force. He ceased to be Anakin Skywalker
and became Darth Vader. When that happened,
the good man who was your father was
destroyed. So what I have told you was
true... from a certain point of view.
That being said, there's absolutely no indication that Luke doesn't already know this name and it would be very odd if he hadn't known it from childhood.
You may be fooled into thinking that there was an earlier reference in ESB...
Emperor: We have a new enemy. The young Rebel who destroyed the Death Star. I have no doubt this boy is the offspring of Anakin
... however this was retconned into the Special Edition (the original wording was "...son of Skywalker")
For completeness, I should mention that the name itself appears in one of the very earliest draft scripts for the movie "The Star Wars", written some 4 years before A New Hope was ever filmed, so clearly Lucas always knew what Vader's real name was, even if the rest of us didn't:
"A harsh gale blows across the bleak grey surface of the Fourth Moon. The leaden sky presses down on a lone figure, ANNIKIN STARKILLER"