What does a US citizen need to do to become a citizen in Iceland?

Try marrying someone from Iceland and then apply for a citizenship. Citizenship

As a general rule foreign nationals must have been domiciled in Iceland for seven years before they can apply for citizenship. Nationals of the Nordic countries, however, may apply after 5 years of domicile. A foreign national who marries an Icelandic national may apply for citizenship after having been resident in Iceland for three years after the marriage. A foreign national, who is cohabiting with an Icelandic citizen, both being unmarried, may apply for Icelandic citizenship if the cohabitation has lasted for more than five years. Application forms for Icelandic citizenship are available from the Ministry of Justice.