Jonathan Creek/WMG

Adam Klaus is British

Adam speaks in a strong American accent, but his sister is British. Adam is such a phony showbiz plank, however, that he's committed himself to faking his American accent around people who know perfectly well he's British, such as his family. The reason they don't call him on it is because they're used to him.

It helps that the first actor who portrayed him was a Fake American.

Adam Klaus is a method actor on a long-running TV series

This follows naturally from the above guess.

Giles on undercover mission

Also following from the above, the "first actor" was actually Rupert Giles on an undercover mission. He was using a glamour spell similar to what happened to Willow when she appeared like Warren, although obviously this was something Giles did to himself. After hearing about Creek's awesome illusions Giles was sent undercover by the Watchers Council to discover if he had actual magic ability or was otherwise supernatural. Once his mission was complete, he left, possibly being assigned to Sunnydale at this time and the real Adam Klaus was released and his memory of the Council kidnapping him removed via magic. We the viewers saw Giles pretending to be Klaus, the other characters just saw the second actor because of the spell.