While most court proceedings are on the public record, courts can order some documents or proceedings sealed. Courts usually have broad discretion in deciding what records can or cannot be sealed, and have similar discretion in deciding to unseal any records. While it may not be possible to have all sealed court records made public, you can petition a court that has ordered those records to be sealed.
Research the law. Courts seal records for various reasons. For example, juvenile cases are typically sealed and cannot be viewed by any member of the public. Depending on the nature of the sealed records, you'll have to determine whether or not the law allows for such records to be made public.
Draft a petition. Once you've established the legal basis for your claim, you must draft a petition asking the court to unseal the records. Depending on the jurisdiction you're in and the court you are petitioning, the form of this document can differ significantly. Contact the court clerk's office and ask if they have any materials you can use, such as templates or packets. The court you're petitioning may keep such materials with the clerk's office, law library or other area, but the clerk should be able to direct you to the proper place.
File the petition. Once your petition has been drafted, you'll need to file it with the court clerk's office. You'll probably have to pay a filing fee, so contact the clerk before you file so you have enough money with you when you are ready. If you are acting as a pro se litigant---meaning you don't have an attorney---you can still do everything yourself, but you may have to comply with other requirements such as filing a pro se notice with the court.
Schedule the hearing. After filing your petition, the court clerk will docket your case for hearing. You'll need to attend this hearing to argue your case before the judge.