Well, the best example of an actor going to Washington would be Ronald Reagan. He made a career of being an actor first, but moved into politics and became governor of California for two 4-year terms, being elected in 1966 and again in 1970, and served from 1967 to 1975. But in regards to your question, he became the 40th President of the United States, being elected in 1980 and 1984, and served two terms as president from January 1981 to January 1989.
Other actors have made it to Washington, to serve in Congress. This goes back to 1944 when a Broadway and movie actress of the 1920s and 30s, Helen Gahagan Douglas, was elected to 3 terms in the House of Representatives. Fred Grandy, from the TV show "The Love Boat," was a 4-term Republican congressman from Iowa beginning in 1986.
Actor/musician Sonny Bono (of Sonny and Cher fame) was a California congressman beginning in 1994. He died in a skiing accident during his second term. And actor Fred Thompson, who appeared on the TV show "Law and Order," served as a senator from 1994 to 2003.
Although Arnold Schwarzenegger is the current governor of California, wrestler Jesse Ventura was the governor of Minnesota from 1999 to 2003, and Clint Eastwood served as mayor of the California city of Carmel, these positions don't officially serve in Washington.