You and your spouse will need to file form that says you meet the conditions that qualify you for a non-adversarial divorce. The paper must be notarized, which means you will need to swear that the information in it is true and sign it in front of the court clerk.
The Judicial Branch has a checklist with all the forms you will need to submit. It can be found here: http://www.jud.ct.gov/webforms/forms/FM248.pdf. If you need help understanding what is on this list or filling out the forms, you can go to a Court Service Center, located at most courthouses.