The Public Defender's Office only handles criminal cases in criminal court. The best way to litigate any improper action by a judge is through appellate attorneys who represent clients in the appellate courts. However, an appellate attorney can only successfully appeal your case only if the judge, the District Attorney or even your own attorney did something significantly wrong.
A defendant who has been convicted after a trial always have the right to appeal the conviction. This process is initiated by the trial attorney. Your attorney will file a notice of appeal in the trial court on your behalf if you wish to appeal your case. An attorney who does not work for the Public Defender, and who specializes in appeals, will then be appointed by the Court of Appeal to represent you on appeal. That attorney will be the one who to consider what you can do about any complaints against any judge who you feel treated you badly during your criminal case.