The Presidential Scholarship is Clark University's largest merit award. It covers full tuition along with on-campus housing and meals for all four years, and it goes to approximately five students a year, picked for academic achievement and leadership potential from the top of the admitted pool.
Nothing extra is required of you. The scholarship committee reads every admitted student's admission application, finalists are invited to interview, and the decision lands with the admission decision. Family finances play no part in who wins it.
For international applicants the relevant background is that Clark awards merit scholarships to a select number of international first-year and transfer students each year, and all applicants are automatically considered for merit aid. Need-based help is a separate track through the CSS Profile, and Clark is need-aware, meaning an application can be declined on financial grounds if the need shown exceeds what Clark can fund.
Clark publishes no separate scholarship deadline, so the admission deadline you choose is the one that matters.