The Kohane lineage goes back over 3,000 years. Among Jews the Kohanes have an elevated community status, but also have some lifestyle restrictions. For at least two thousand years communal leaders have exercised caution when accepting the Kohanic status of community members. Nonetheless, there certainly have been cases where people were accepted as being of the lineage who were not – thousands of years is a lot of generations, a lot of migrations, and errors must have happened. Nonetheless there is good reason to believe a meaningful percentage of the people who think they are Kohanes.
The standard “proof” given is grave markers and written contracts. By tradition, gravestones for Kohanim include the Kohanic status in the name of the person buried there, and often in the form of an engraved image of Kohanic hands.
In my case the main “proof” of my Kohanic status is my father’s grave marker. His name in Hebrew (and Yiddish) notes that his is a Kohane, and below his name the image of two hands shows those hands in the pose of a Kohanic Blessing.