How does Charlie find out that Edward Cullen is a vampire?

Charlie Swan does not find out that Edward Cullen, or the Cullen family are vampires. He does however know that they are not strictly human and are supernatural, just as Jacob is a shape-shifter.

Charlie discovered this change when Jacob told him that in order for Bella to get well (after the Cullens had told him she was ill) was to become like Esme and Alice. He then revealed himself to be a shape-shifter by changing into his wolf form as proof that supernatural creatures exist.