According to U.S. Constitution Online,
The authority of the Congress to conduct the census in whatever way it wishes, and thus to require that the forms be filled out is found in the Constitution itself, which notes:
[The Census] shall be made ... in such Manner as [Congress] shall by Law direct.
The Congress is also authorized to ask various questions in the census aside from the basic headcount by virtue of this clause and by virtue of the Necessary and Proper Clause.I believe many of these questions to be neither necessary nor proper. What does the government need with information about the home you live in and whether or not you have a flush toilet?
See the form for other even more intrusive questions. What's next? "How often does each person in this household have sex"?