Depending on the severity, you can become a permanent US citizen with a criminal record. A criminal record does make the process harder, and USCIS does not guarantee anyone the right to earn a green card.
See below to learn more about earning a green card with a criminal record. This article is not a guide; you should not rely on it for legal advice. It only provides a very brief overview of what you might expect.
The US Citizenship and Immigration Services runs a background check for every green card applicant. The process includes a fingerprint check and an FBI name check. Even though the USCIS runs a background check, providing criminal history information is still in your best interest. Even if you have an arrest on your record without a criminal charge, you should include this on your application.
Some crimes make you ineligible for green card status. The US takes a hard stance against drug crimes. You may receive a drug crime waiver for possessing 30 grams or less of marijuana, but any other drug crime makes it difficult to earn a permanent residency.
Crimes of moral turpitude also disqualify you from permanent residency. The USCIS considers acts such as rape, fraud, animal abuse and homicide as crimes of depraved or immoral behavior.
Finally, aggravated felonies make you inadmissible for green card status. The list of aggravated felonies changes, but they usually involve violence or drug crimes.
The United States does not prevent everyone with a criminal history from moving into the country. However, crime certainly does not help your case. You must demonstrate you are a reformed and productive member of society before earning your green card.