In what TV show did Leonardo DiCaprio make his first appearance?
DiCaprio made his television debut as Glen in a couple of episode of The New Lassie in 1989.
No, he wasn't even nominated. Leonardo DiCaprio has been nominated for Oscar three times. He has not yet won. Nominated Oscar Best Performance by an Actor in a Leading Role for: Blood Diamond (2006) Nominated Oscar Best Performance by an Actor in a Leading Role for: The Aviator (2004) Nominated Oscar Best Actor in a Supporting Role for: What's Eating Gilbert Grape (1993) your question does not make sense.
Leonardo diCaprio did not star in a "TV version" of any Shakespeare play. He starred in Baz Luhrmann's 1996 film Romeo+Juliet, opposite Claire Danes, which was released in theatres long before it was ever seen on TV. Romeo and Juliet has been produced at least five times specifically for TV, so it doesn't make sense to talk about "the TV version" of this play.