What is the Doctor's real name? No-one knows, and it would spoil the series if we did know. This is what the title of the show implies. Doctor "Who"? Some things need to stay mysterious for the show to work well. When the Doctor meets someone, he always introduces himself as "the Doctor" (note the capital "D"). He has sometimes used aliases, such as "John Smith", but it is clear that these aliases are not his real name. Even his companions don't know his real name. Possibly only River Song knows his real name.
The Doctor doesn't generally go around announcing that he's an alien. He only seems to do it to remind his human companions that he is not human. If I remember correctly, the First Doctor told Ian and Barbara in the first story that both he and Susan were born on another planet (Gallifrey was not named until much later in the series).
Some UNIT personnel were aware that the Doctor isn't human, notably the Brigadier eventually accepting that the Third Doctor was the same person as the Second Doctor, as well as actually seeing the Third Doctor regenerate into the Fourth Doctor. The Fourth Doctor also told Sarah Jane Smith that he was an alien and not to expect him to react the same as a human would.
The Sixth Doctor tried to brush off his rudeness and bad behaviour as being alien and Peri compared him to the Fifth Doctor, who generally behaved much more human than other Doctors.
In the new series, the Ninth Doctor told Rose that he's an alien as part of his invitation to her into the TARDIS. Each companion has been aware that the Doctor is an alien, often demonstrating this by checking both of his hearts, but not all of them have been told this on-screen.
The Doctor is also quite capable of hiding that he's an alien. There have been several instances where someone has called the Doctor and his companions human, and apart from reacting slightly (for the benefit of the audience), the Doctor doesn't disabuse the other person of their assumption.
Sometimes the Doctor likes to boast about how old he is, or even that he's so old he can no longer remember his exact age. When it comes down to it, the Doctor is a very complex person who doesn't often react the same way. He can show off or be modest, he can state with great confidence how old he is and at other times have no idea. The Doctor is not always self-consistent.
And remember: The Doctor lies.