The Naturalization Requirements are listed on the U.S. Citizenship and Immigration Services:

http://www.uscis.gov/portal/site/uscis/menuitem.5af9bb95919f35e66f614176543f6d1a/?vgnextoid=12e596981298d010VgnVCM10000048f3d6a1RCRD

Generally, you must show that you have been a person of good moral character for 5 years in order to qualify for naturalization/citizenship.  You need to establish good moral character over the past five years. Sometimes they can go back beyond the five year period.

Good Moral Character
Generally, an applicant must show that he or she has been a person of good moral character for the statutory period (typically five years or three years if married to a U.S. citizen or one year for Armed Forces expedite) prior to filing for naturalization. The Service is not limited to the statutory period in determining whether an applicant has established good moral character. An applicant is permanently barred from naturalization if he or she has ever been convicted of murder. An applicant is also permanently barred from naturalization if he or she has been convicted of an aggravated felony as defined in section 101(a)(43) of the Act on or after November 29, 1990. A person also cannot be found to be a person of good moral character if during the last five years he or she:

• has committed and been convicted of one or more crimes involving moral turpitude
• has committed and been convicted of 2 or more offenses for which the total sentence imposed was 5 years or more
• has committed and been convicted of any controlled substance law, except for a single offense of simple possession of 30 grams or less of marijuana
• has been confined to a penal institution during the statutory period, as a result of a conviction, for an aggregate period of 180 days or more
• has committed and been convicted of two or more gambling offenses
• is or has earned his or her principal income from illegal gambling
• is or has been involved in prostitution or commercialized vice
• is or has been involved in smuggling illegal aliens into the United States
• is or has been a habitual drunkard
• is practicing or has practiced polygamy
• has willfully failed or refused to support dependents
• has given false testimony, under oath, in order to receive a benefit under the Immigration and Nationality Act.

An applicant must disclose all relevant facts to the Service, including his or her entire criminal history, regardless of whether the criminal history disqualifies the applicant under the enumerated provisions.