Adam Clarke on Acts 25:10

I stand at Caesar's judgment seat - Every procurator represented the person of the emperor in the province over which he presided; and, as the seat of government was at Caesarea, and Paul was now before the tribunal on which the emperor's representative sat, he could say, with the strictest propriety, that he stood before Caesar's judgment seat, where, as a freeman of Rome, he should be tried.

As thou very well knowest - The record of this trial before Felix was undoubtedly left for the inspection of Festus; for, as he left the prisoner to his successor, he must also leave the charges against him, and the trial which he had undergone. Besides, Festus must be assured of his innocence, from the trial through which he had just now passed.