Your aid package will not depend on what your family is willing to pay, but what the college/university thinks it is able to pay. Be sure to have some institutions that are fully affordable on your list. If your family cannot afford anything here, be sure to apply to some places in your home country or in other countries that are affordable.
Occasionally, students are able to negotiate better financial aid packages, but in those cases the students have been offered different amounts of aid by colleges/universities that are peer institutions. In other words, if FamousA offers $X and FamousB offers $X-Y, sometimes FamousB might be willing to come up with slightly more money. However, not all colleges/universities do reconsider aid packages.
In order to get your student visa, you will need to be able to demonstrate that you have your full Cost of Attendance (as defined by the college/university itself) covered. This money can come from the college/university in the form of financial aid, from your family, or from another source. Many colleges/universities simply do not admit international students who they believe cannot afford to attend.