Wow, excellent question, but a broad one.
Here are a few suggestions:
a. Show a consistent pattern of academic achievement through high school. Nail SATs/ACTs--study for them, prepare for them, and do your absolute best.
b. Demonstrate involvement and leadership in a variety of different extracurricular activities, jobs, community service, athletics, etc. Notre Dame wants well-rounded people to attend.
c. In your essays, demonstrate a strong command of the English language, the ability to view an issue from multiple points of view, and the ability to choose a point of view and explain it clearly. In your personal essay, make it truly personal. Let Notre Dame see the person they would be accepting.
d. Be very choiceful about choice of references/letters of recommendation. You want a wide variety of people who know you from different classes or activities. Make sure it is someone who knows you well enough to write a great recommendation for you that would not be relevant to someone else (IOW, not a "form letter" recommendation).
e. It never hurts to set up a phone "informational interview" with an admissions staff member or the director. This gives you a chance to get straight from the source the kinds of things they look at in determining who will go to Notre Dame.
f. Contact the local Notre Dame alumni club to see if they have HS student events, or things like that.
In the end, Notre Dame will accept someone who they believe will give to the ND community as much as he/she will receive in a ND education and degree.
Good luck!