Save values generated via API as taxonomy terms

It’s as easy as:

function wpse_199523_save_imdb_terms( $post_id ) {
    if ( ! $imdb = get_field( 'imdb_id', $post_id ) )
        return;
    if ( ! $data = imdb_connector_get_movie( $imdb ) )
        return;

    if ( ! empty( $data['actors'] ) )
        wp_set_object_terms( $post_id, $data['actors'], 'actors' );
    if ( ! empty( $data['director'] ) )
        wp_set_object_terms( $post_id, $data['director'], 'directors' );
}

add_action( 'wp_insert_post', 'wpse_199523_save_imdb_terms', 500 /* Ensure we run after ACF */ );

Refer to the codex on wp_set_object_terms