Short Answer: The Doctor was born in what we would perceive of as the past. If he is correct about his age (900 Earth years) and he was present at the The Last Great Time War which is in the 26th Century AD, Earth Time, he would be a time-traveler who was born in the 1600's AD Earth time. It is difficult to pinpoint his actual location in time due to the differential between the normal flow of time and the compressed Inner Time of the planet Gallifrey.

He would have lived through the periods of the 20th and 21st Century and eventually lived long enough to be at the final battle at Arcadia in the 26th century. He then, returned to Earth nostalgically adventuring through the 19, 20 and 21 century trying to console himself of a time before the final fall of the Gallifreyans.

Complex Answer: The Gallifreyans evolved just like any other race in the Universe. They did not simply spring into being, fully formed. They use technology and likely developed it just like any other species would and later achieved a level of technological supremacy after millions of years of development. This can be supported by the existence of such technology as the TARDIS itself. As such, it can be supported that despite their mastery of time, they would have come from the past, as we are able to see and understand the passage of time.

The Gallifreyans developed the capability to time travel after they created the Eye of Harmony, estimated to have come into existence 3.5 billion years ago, Earth Time. (from the Doctor Who novelizations.) Once the Eye of Harmony was created, the Gallifreyans removed themselves from the time stream proper and began to exist outside of time, a thing they called Inner Time (a temporally delayed, entropy free zone outside of the normal flow of time).

The Eye of Harmony acted as the focus and beacon for all TARDIS technology and was later destroyed by the Doctor during the Last Great Time War. He cast the Eye of Harmony into the Gallifreyan sun to destroy the fleet of one million Dalek ships and the City of the Gallifreyans. The destruction of the Eye of Harmony caused a temporal shockwave all the way back to the Big Bang, timelocking the event forever. The energy of the Eye was linked to the time stream energy and any remaining TARDIS uses temporal energy to travel through time now that the Eye is gone.

Despite the Doctor's ability to move through time with what would appear to be apparent ease, there are places in time and space he is still unable to go, implying there is time and space the TARDIS is not able to manipulate.

The Void between Universes

Beyond a limited distance outside of our universe

The temporal bubble imprisoning the Gallefreyans

Any moment in time he has already existed in, preventing temporal paradox

Fixed points in time that are not able to be changed (as described by him)

Since he has blocked the Gallifreyans from the contemporary universe including any timelines that would allow them to exist in parallel universes, they have no future and for them to be effectively isolated, they can have no past or access to the past. If they did have access to the timestream, they would be able to escape their temporal prison, through the past.

As for the appearance of the Doctor, at any particular moment in space and time, he could arrive at any point not forbidden to him from either the past or the future.

Here is an infographic of adventures of the Doctor since the series first started. You can reach the full sized graphic here.