Figure out your financing. A biplane is rarely cheap; khow you will finance the plane.
Decide what you want the biplane for. Some people buy biplanes for museum reasons: the planes are a part of history and make wonderful historical artifacts. Others, though, actively fly and maintain biplanes.
Consider new biplanes. There are lots of newly-built biplanes that are modeled after the classics. These planes, like the 1981 EAA Bi-Plane P2, for example, are less expensive and as reliable as a ground-up restored classic biplane.
Contact a classic plane dealer. Companies like Old Thyme Aviation are expert as selling restored biplanes. Speaking with a dealer like this will help you get a good context and some expert advice on buying a biplane.
Have the plane surveyed. A professional surveyor will be able to spot problems, flaws and potential dangers in a biplane and save you lots of trouble and money in advance.