Why doesn't the Doctor consider himself a hero?

Atenção: May contain spoilers. (Mainly from the rebooted series.)

From an external perspective he is typically considered to be either a hero or anti-hero depending on the incarnation and which acts one is considering.

The Doctor himself considers himself to not be a hero for many reasons.

Firstly because he (almost) never seeks situations which would put him in a position to perform heroic acts. Contrast this to 'traditional' heroes who often make it their job to be heroic, for example warriors such as Heracles and Achilles, or more 'everyday heroes' such as firefighters, police and (ironically) doctors. Instead, the Doctor simply travels and trouble usually finds him.

During his travels, the Doctor committed a great many crimes and morally questionable acts, many of which we still don't know the full extent of (e.g. the events of the time war). Aside from the numerous murders he's committed, he's been a guilty party in a number of genocides:

• The Tenth Doctor went back on his fourth incarnation's 'have I the right?' speech and outright murdered the entirety of the Racnoss empress' children, thus making him accessory to genocide (if not the main perpetrator) when the British army wiped out the empress herself. (The Runaway Bride)
• The Fourth Doctor destroyed Sutekh, the last known Osiran. (The Pyramids Of Mars)
• With Leela's help, the Fourth Doctor destroyed the last of the Fendahl. He dropped the remains of the Fendahl into a supernova to ensure their permanent destruction. (Image Of The Fendahl)
• The Fourth Doctor killed the last of the Great Vampires and the only vampire survivor of the Vampire-Time Lord war. (State Of Decay)
• The Sixth Doctor wiped out the newly created Vervoid race on Hyperion III. A crime for which he was actually tried in court. (Terror Of The Vervoids)
• The Eleventh Doctor was responsible for wiping out the last of the Saturnyns. The last one to die stated that she hoped the species' extinction would haunt his conscience. (The Vampires Of Venice)

Some miscellaneous examples of the many (albeit less serious) crimes has committed over the years:

• The First Doctor stole the Tardis he now travels in (it is still stolen property, the Doctor never returned it or obtained lawful ownership).
• The First Doctor kidnapped Ian Chesterton and Barbara Wright. (An Unearthly Child)
• The Eleventh Doctor was prepared to lobotomise the last star whale in order to save a single ship of humans (a much overpopulated species). (The Beast Below)
• The Eleventh Doctor stole the Byzantium's Home Box from the Delirium Archive. (The Time Of Angels)
• The Twelfth Doctor has broken into the bank of Karabraxos and stolen a memory chip and a gene suppressant (and arguably a slave if one considers a slave to be the property of a person). (Time Heist)

On top of which he certainly has a mean streak.

• The First Doctor was perfectly happy to deceive his captive audience and his granddaughter for the sake of investigating the Dalek city on Skaro, endangering them all in the process. (The Daleks)
• The twelfth doctor has forced his companion to coax humanity into deciding whether or not to kill an unborn creature to save themselves, all whilst knowing full-well that choosing to not kill it would cause no harm to the earth. (Kill The Moon)
• The Twelfth Doctor has also had various violent arguments with Danny Pink over time, in which he hammered home his prejudice against soldiers.
• O 'Timelord Victorious'. Enough Said. (The Waters Of Mars)

The Doctor remembers the times he's done wrong and the times he's saved lives, and he knows that he doesn't do the things he does to be heroic. He's just a time traveller trying to help out where he can, and he doesn't always get it right.

I dare say if he had the choice he would rather the majority of his trips featured as little peril as possible, like the space tourist he longs to be. Being able to visit a nice hotel without its architect going mad and its inhabitants turning on each other; being able to take a ride on the last ship out of earth without ending up in a whale's mouth; being able to take a trip to a planet made of diamond without encountering a strange mimicking entity; and being able to go to a space theme park without it being overrun by cybermen. Like how Sherlock Holmes retired and took up bee keeping or how Winston Churchill loved to tend to his garden. This was very much alluded to when the Doctor became Victorian teacher John Smith, before being forced to give it up to resume his true persona since he was the only one with the knowledge to save everyone. (The Family Of Blood)

Finally, he's hardly first person to be heroic and then vehemently protest that they are not a hero. Many veterans of various wars have refused to accept their medals because they believed they were just doing their duty, no matter how important what they achieved was or how deserved the accolade might be. There are also various literary characters who have similar feelings about their perceived heroism.

It varies. Sometimes it is because he views himself as ordinary; sometimes because he views himself as a bad person.

Because he has committed atrocities

The Doctor believes himself to have killed all the Dalek and Time Lords. While one could argue that the Daleks had it coming to them, the Time Lords may not have. In any case, the destruction of an entire species, even a wholly evil and xenophobic one, is nothing to take lightly.

He also destroyed all the young of the Racnoss, and wiped out countless other species. Often in self-defense, yes, and often to save countless more, but nonetheless.

He left Rose in an alternate reality.

He wiped Donna's memories (albeit to save her).

He has caused the deaths of some of his companions (Peri, Clara, Kamelion, Adric), and caused others to be changed beyond recognition (Jack, Donna), or stranded far from home (Amy, Rory, Rose).

This is enough to make anyone believe that they aren't a hero, até if they have done just as many things to help others.

Because he views himself (in his better moments) as an ordinary person

"I am not a good man! I'm not a bad man. I'm not a hero. I'm definitely not a president, and no, I'm not an officer. You know who I am? I… am… an idiot! With a box and a screwdriver. Passing through. Helping out. Learning. I don't need an army. I never have. Because I've got them, always them, because love is not an emotion. Love is a promise."

Death in Heaven [8.12]

He believes (at this point, anyway), that he is no better than other any person.

True, he is a Time Lord, and possesses talents beyond most other creatures in the universe. But then, so does a human, at least relative our current reference pool! Yet we do not wake up every morning feeling like extraordinary sentient beings. The Doctor has plenty of individuals relative to which he can feel inadequate.

Because he is not a hero, he is a Doctor - which needs a bit of Hero in him, of course, but is not a Hero per se.

In O dia do médico, towards the end when the three doctors are looking at the switch that will blow up Gallifrey, Matt Smith looks at Clara and doesn't know what to do.

She reminds him that he has been a Warrior (John Hurt) and a Hero (David Tennant), but what the world actually needs is him: a Doctor. A doctor doesn't save lives for the sake of being a Hero; a doctor saves lives because that's what a doctor does.

Metaphorically, she's describing each of those predecessors as really a part of himself - an element of his character. Doctors must be heroic (by saving the lives of those around them), and warriors (fighting to save those lives). But they also are more than just that - they make life or death decisions intelligently, sometimes choosing something that is not necessarily obviously "right".

Saving lives is what the Doctor does, after all - when he steps in and stops various plots from occurring, it's always for the purpose of saving lives. He doesn't act as a Hero, looking for a bad guy to vanquish, and always Doing Good; he acts as a Doctor, looking for lives to save, and how to save the most lives - even at the cost of doing perhaps-bad things.

I feel that this is often a major part of the character development for each doctor, coming to grip with the fact that he is not a Hero, and that trying to be one often gets him into trouble. Realizing that he has limitations, but that he's also a sum greater than his parts, is how he becomes the Doctor in truth - and lets him make the decisions he must make in order to save the world.