Renouncing your U.S. citizenship, if you have a dual nationality, is a simple process, but one that must be undertaken outside of the United States. The process is described in Section 349 (a) (5) of the Immigration and Nationality Act, as that Act is embodied in 8 USC 1481 (a) (5), as, "making a formal renunciation of nationality before a diplomatic or consular officer of the United States in a foreign state, in such form as may be prescribed by the Secretary of State." Other actions, called "expatriating acts," may also rid you of your U.S. citizenship.
Leave the United States. The law is specific on this point, that the renunciation of citizenship must be made "in a foreign state."
Make an appointment with a U.S. Embassy officer upon arrival in a foreign country. The U.S. Department of State website says that you must, "Appear before a U.S. consular or diplomatic officer in a foreign country, normally at a U.S. Embassy or Consulate."
Sign an Oath of Renunciation freely and voluntarily. The State Department website is specific on this point: "Renunciations that do not meet the conditions described above have no legal effect. Because of the provisions of section 349(a)(5), Americans cannot effectively renounce their citizenship by mail, through an agent, or while in the United States. In fact, U.S. courts have held certain attempts to renounce U.S. citizenship to be ineffective on a variety of grounds."
Freely and voluntarily accept a commission as an officer in the armed forces of another country, or serve as a non-commissioned officer in those armed forces. This is called an expatriating act.
If you voluntarily serve in any capacity in the armed forces of a country "engaged in hostilities" with the U.S., you also forfeit your U.S. citizenship.
Take an oath of allegiance to a foreign government. Such an oath is an expatriating act, as is seeking naturalization as a citizen of a foreign government.
Commit an act of treason. A conviction for treason may be the ultimate method of ridding yourself of your American citizenship. No U.S. citizen has been put to death for the crime of treason, even though treason is punishable by death: before a traitor is executed, he is formally stripped of U.S. citizenship.