While it is possible to claim dual citizenship, meaning a person holds citizenship in two countries (sometimes three), the United States being one of them, the legal process for doing so is long and complicated and may leave you to question if the resulting effort is worth it. For the record, there is no official application or legal process in which to apply for "U.S. dual citizenship," on the other hand, the U.S. State Department does recognize dual citizenship.
Obtain U.S. citizenship by either being born in the United States or in another country to legal citizens of the U.S (which automatically makes you a dual citizen). You can also become a U.S. citizen by obtaining a Green Card and going through the process of becoming a naturalized citizen of the U.S. Taking this route means you will have to surrender your passport and renounce citizenship to your home country.
Contact the consulate for your home country, either by telephone or Internet, once you have become a U.S. citizen.
Ask the consulate if you will be allowed to obtain a passport from your home country. Some countries will permit you to obtain a passport without having to renounce your newly acquired citizenship from the United States, while other countries, with more extreme governments, might punish you with a prison sentence (should you decide to return).
Follow the steps outlined by the consulate to obtain a passport from your home country. Once you obtain the passport, you will once again be considered a citizen of that country---as long as it does not require you to renounce your U.S. citizenship. This process will automatically make you a dual citizen.
Note there is no official policy in the United States that says you cannot be issued a passport from another country if you are citizen of the U.S. Furthermore, the U.S. government will not revoke your citizenship if you claim citizenship in another country. If you've done your homework, and have basically kept political opinions to yourself, your goal to achieve dual citizenship through legal means might become a reality.