Would actually be quite good if done right and not just using him as a mouth piece to spout vague progressive platitudes.
Massive survivors guilt, may have been responsible for his own people's extinction although in truth they would have been dead in a few hours anyway and at least this way they took their attackers out with them and he spends the better part of a millennium becoming increasingly unhinged as the loneliness eats at him. Occasionally something will trigger his damage and he goes fucking ballistic and not the lovable maverick type of ballistic, oh no. The unapologetic genocide of a species ballistic. People know what it is to be normal by reference to others of their kind around them. Doctor has nobody but his own self for company as he gradually becomes more toxic and self destructive. Has to keep enticing people to come with him into the inter-dimensional creepy transit van to act as a surrogate morality. He want's to be good (it's questionable if he even remembers what that means ore meant to him), but he has no moral compass so he takes people who do and gauges his actions on how happy/unhappy what he is doing makes them.
It's why he can't stay still for very long. If he stops to think he starts to see what he has become and he can't live with it. His grad declarations of such self evident bullshit like "I'm The Doctor and I help people!" are to convince himself and people who don't know him and look good in front of the "companikons" because he likes to look good as it makes him feel better about himself. His fascination with Earth and protecting Earth is because he likes their adoration and he gets pissy when we start to stand up on our own because then we won't need him and then who will praise him?
Chaotic Neutral (at best) with delusions of Chaotic Good sorcerer + bard.