Goth had a brother named Rath.
On behest of the CIA, Goth masqueraded as a fictional character Lemuel Gulliver in the Land of Fiction, in order to monitor the Doctor. Shortly afterwards, on Gallifrey, Goth with two other Time Lords, would preside over the trial of the Doctor, which would decide the latter's exile on Earth. Goth was only a Councillor at the time.
It was Goth who persaded the CIA to send the Doctor to Peladon.
He travelled to the planet Tersurus and found the dying Master there. Aware by now that the Lord President intended to name Cardinal Borusa as his successor rather than Goth, the Chancellor entered into an uneasy alliance with the Master, who promised him the presidency in exchange for assisting with killing the Doctor. Goth subsequently assassinated the President (while setting up the Doctor as an unwitting decoy assassin) before he could announce Borusa as the new President, resulting in what should be been a largely unopposed election for the President's office. However, the Doctor announced his own candidacy, so as to buy himself time to investigate.
Inside the Matrix, he fought the Doctor, initially using his knowledge of the Matrix's workings to his advantage. However, the Doctor soon gained the upper hand, and the infuriated Master tried to overload the Matrix by sending a massive energy spike through Goth's brain. This failed to kill the Doctor, who escaped from the Matrix in time, but fatally damaged Goth's brain and destroyed his regeneration cycle. The Doctor found Goth and the seemingly dead (but actually self-sedated) Master afterwards, and the dying Chancellor related to the Doctor how he and the Master had come to be allies, and why he wanted to kill the previous President. Goth died shortly afterwards, and it was decided by Borusa, ironically the man who would have taken the Presidency from him, to cover up the actual truth of his death and to tell the population that he had sacrificed his life to kill the Master.