The bulk of people working in international affairs in Washington work for agencies other than the State Department. Unfortunately there is no single recruiting device such as the Foreign Service exam for these organizations. The larges employment opportunities are the Defense Department (both military and civilian) and the intelligence organizations, particularly the Central Intelligence Agency and the National Security Agency. Information about military careers can be obtained from the ROTC groups on campus. Civilians hired by the Defense Department tend to be people with particular specialties; advanced degrees are usually required. Given the informal hiring process, actual job experience, which in practice means internships, is very important.
Intelligence careers can be divided into analysts (people who work with secret material trying to decide its significance) and clandestine operators. Anyone interested in such positions should look at the book Careers in Secret Intelligence by David Atlee Phillips, a former CIA officer; David Wise's "Campus Recruiting and the CIA," New York Times Magazine, June 8, 1986 is also useful. The Central Intelligence Agency and the National Security Agency both hire junior-level career people on the basis of exams. Contact each agency separately to see what their current needs and procedures are. They also hire many people with particular skills for analysis, usually with advanced degrees. They seem to be particularly interested in exotic languages, geographic area specializations, economics, political science, international relations, mathematics, computer science, engineering, and physical science. Again internships are particularly useful here.
The Agency for International Development (AID) administers American foreign aid and has a fairly large staff. In general it seems to recruit people with technical training in areas like economics or agriculture. Their relationship with the State Department changes with each reorganization; if you are interested, contact them directly. Smaller organizations include the Export-Import Bank and the Office of the Special Trade Representative.
Many "domestic" executive agencies have international activities or offices; these are often small, but sometimes they offer interesting opportunities. Commerce, for example, is concerned with foreign trade, Agriculture with farm exports, Justice with international legal issues, etc.
The number of people on Congressional staffs concerned with international affairs has greatly increased in the past few years. There is no single recruiting process for such jobs; people are selected on the basis of contacts, past experience, and educational qualifications, roughly in that order. Internships are crucial for anyone interested in these sorts of positions.