Tony Mauro had an article in last week's National Law Journal titled Colleagues Recall Chief Justice's Early Days about the appellate career of Chief Justice John Roberts who was "widely viewed as one of the best Supreme Court advocates of his time." His "record" was 25 wins in 39 SCOTUS cases between 1989 and 2003, but "among connoisseurs of Supreme Court advocacy, it's not only winning that counts. It is how you prepare, the quality of your briefs and how you hold up under intense questioning at oral argument" -- areas where Roberts' career "has taken on mythic proportions" because he prepared "obsessively."
Tips to consider emulating: engage in multiple moot courts; visit the actual locations involved in your appeal to better understand context and the facts on the ground; prepare index cards with your key points -- and shuffle them so you can make smooth transitions from point to point in any order; as respondent/appellee, pay close attention to the argument and begin your argument where the other side left off and/or where the justices left off (rather than using pre-prepared remarks that don't continue the conversation of what just happened).
The Chief declined to be interviewed for the article, but several pithy quotes from him from other sources were included, including this: "My reputation as a Supreme Court and appellate advocate took an enormous leap when I became chief justice. I had no idea I was as good as people tell me."