First off, being impeached just means being formally accused. Any federal judge (including a Supreme Court justice) who is impeached would be entitled to a trial in the Senate, and would only be removed from office if the Senate votes to remove him after such a trial.

If the chief justice is successfully impeached, the Court will function just as it would if he had resigned abruptly or died. The senior associate justice (currently Antonin Scalia) would serve as acting chief justice, and the president would nominate a successor. Once a successor was confirmed and sworn in, he or she would become the new chief justice.