Nearly all private international law problems will involve researching the specific laws of both the U.S. (assuming a U.S. citizen or company is part of the legal scenario at hand) and one or more foreign countries. This is because, as stated before, there is not one well-defined body of private international law; rather, each country has its own way of handling private international law/conflict of laws issues. You will want to find out a country's rules about jurisdiction, choice of law, and/or enforcement of judgments depending on the facts of the case.
One useful starting point is Reynolds & Flores Foreign Law Guide:
For more information see the UCLA Law Library's guide to foreign law research: