The Two

The Two was the main alias used by the second incarnation of a Renegade Time Lord who suffered from regenerative dissonance.

Trying to stop his future
After regenerating for the first time, the Two found out he suffered from regenerative dissonance. He travelled to Karn to consult the Sisterhood and know if he could avoid his fate. He instead found out that the Doctor would repeatedly thwart his future attempts and be in his way. Determined to stop him before it happened, he tracked him down to Earth, but the Doctor tricked him and took away his TARDIS, before leaving him to UNIT.

Eventually, though, the Two managed to escape. He deceived Harry Sullivan, making him think he was a future version of the Doctor, into studying a piece of desentherium, a mineral which could open a time corridor. Unknown to him, the War Doctor, having reverted to his sixth body due to an attack from a degeneration weapon, had appeared to investigate his condition. The Doctor resolved the experiment for him and Harry, allowing the Two to travel back to 1963 and attempt to blow up the Doctor's TARDIS with a warp bomb, which would have destroyed it, thus avoiding his future from happening. The Doctor, however, managed to reach his TARDIS and stop him in time by connecting himself to the TARDIS' telepathic circuits and confusing the Two into stupor. He then abandoned the Two on a desert planet to give him a chance at redemption. (AUDIO: Two's Company)

At some point after this, the Two came to serve the Union, a future incarnation of his, who placed him in a cybernetic suit which blocked out the One, but gave the Union complete control over his mind and body. Serving the Union, now under the name Operator Zero, he helped her maintain her Diamond Array, a space station which captured stars and planets and condensed them into diamonds. When the station was being sabotaged by River Song and Susan Foreman, River was able to free the Two from his cybernetics and let him use her vortex manipulator to escape. He bore no memory of his service as Operator Zero once he was freed, nor of the Union. (AUDIO: The Union)

Post-mortem
As the Eight began to regenerate, the Two told him that he tried his best but it was time for a 'better' man to take charge. He also told the Six to be quiet, telling him that none of them get to be in charge anymore, and to concentrate during the Eight's regeneration. (AUDIO: The Doomsday Chronometer)

When the Eleven was imprisoned by the Seventh Doctor, the Two claimed the former got caught by the latter due to overreaching himself and failing to account for all of The Doctor's 'moves'. As the freezing process began, the Two offered to make an arrangement with the Doctor in order to free themselves. (AUDIO: The Eleven)

The Two boasted that his "nose was particularly good at sniffing out traitors." Within the Eleven, he exposed Volstrom's betrayal of Padrac as an agent of Ollistra. (AUDIO: Songs of Love)

Personality
While on a mission for the High Council with the Twelve, the Eighth Doctor described the Two as "the charmer". (AUDIO: Planet of the Ogrons).

He is somewhat rude but ultimately tired man who just wants peace and quiet. He would be harmless if it weren’t for the fact that the One is slowly driving him insane. (AUDIO: Two's Company)

Behind the scenes

 * The script of Dark Universe describes the Two as "calm, reasonable, convincing, charming, liar and manipulator".