There have been a few instances on the show that seem to indicate he was the first. The Third Doctor story "The Three Doctors" (transcript here), in which the Third met up with the Second and First, has the Time Lord President tell an underling to "Show me the earliest Doctor", after which they cut to a shot of William Hartnell's First Doctor. The Fourth Doctor story "The Deadly Assassin" established that Time Lords can only regenerate twelve times (so they can only have thirteen bodies over the course of their lives), and in the Fifth Doctor story "Mawdryn Undead" (transcript here), the Fifth Doctor says "I can only regenerate twelve times. I have already done so four times." So, if he's telling the truth this would indicate the first five Doctors seen on TV really are his first five incarnations.
A little later in the Fifth Doctor's run there was "The Five Doctors", and from the transcript here, there was this exchange where the First Doctor met the Fifth:
SUSAN: Is he really
DOCTOR 1: Me? Yes. Yes, I'm afraid so. Regeneration?
DOCTOR 5: Fourth.
DOCTOR 1: Goodness me! So, there are five of me now!
Shortly after this, the First Doctor introduced himself to the Fifth Doctor's companion Tegan by saying "As it happens, I am the Doctor. The original, you might say."
That said, a possible element of doubt is introduced in the Fourth Doctor story "The Brain of Morbius" (transcript here), in which the Doctor challenges the evil Time Lord Morbius (who has been reduced to a brain controlling a Frankenstein-like body) to a "mind-bending contest", which the Doctor describes to his companion Sarah Jane as "Time Lord wrestling". Once they are strapped into the machine, they seem to have a mental battle which involves pushing back through one another's previous regenerations, and there is a sequence that seems to show versions of the Doctor before William Hartnell's "first":
Fans commonly suggest that the faces shown in this sequence were actually Morbius' earlier regenerations, and it's shot in a somewhat ambiguous way so I think this is a perfectly acceptable retcon given how much other evidence there is from other episodes that William Hartnell's Doctor was really the first. But several people involved in the production of the episode have indicated that their intent was that these faces were earlier regenerations of the Doctor, as discussed on this page:
The answer to which Time Lord these extra faces belongs rests in the
production team. When questioned directly about who owns the faces
writer Terrance Dicks answered, "I have no idea who the faces in the
mind battle were. You would have to ask [co-writer] Bob [Holmes]."
 The answer is instead revealed in an interview with producer
Philip Hinchcliffe who commented on the mindbending sequence, "We
worked out what period we wanted each image of the Doctor." It
was production team members that were dressed in period costumes to
be the Doctor in photographs used for the sequence. He also added
We tried to get famous actors for the faces of the Doctor. But
because no-one [sic] would volunteer, we had to use 'backroom boys.'
And it is true to say that I attempted to imply that William Hartnell
was not the first Doctor.
Hinchcliffe echoed this concept in yet another interview adding,
"There was no subversion of the mythology of the series intended, but
I just reasoned that it was entirely possible that William Hartnell
may not have been the first Doctor Who. So yes, as far as Bob
[Holmes] and I were concerned, the other faces were meant to be past
Doctors." Robert Holmes, the script editor for the show when the
story was made, also supports this as the production teams intention:
'[The Doctor] is mortal, and has only so many regenerations,' he
commented in later years. 'We don't know which one Hartnell was,
whether he was first or not. In the phantasmagoric scene where they
are mind-wrestling, we see the Doctor forced back through a number of
It is quite obvious that when the story was made, the faces on the
screen during the mindbending contest were to represent the Doctor.
The "Myths" section of the wiki article on The Brain of Morbius lists some additional stories other than the ones I mentioned that have indicated that William Hartnell's Doctor was really the first:
The mindbending contest at the end of part four proves conclusively
that the Doctor regenerated multiple times prior to William Hartnell's
Doctor. While this was apparently what the production team intended to
imply, the overwhelming weight of evidence offered throughout the rest
of the series contradicts this. For instance, Time Lord dialogue in
The Three Doctors specifically establishes that the Doctor as
portrayed by William Hartnell is "the earliest Doctor"; further
evidence provided in Mawdryn Undead, The Five Doctors, the 1996 TV
movie, The Next Doctor, The Lodger, The Name of the Doctor, and The
Time of the Doctor) (among others) establishes that all of the
Doctor's incarnations are accounted for in his televised adventures.