Was Hitler exclaiming "Heil Hitler!" himself when saluting?

Short Answer

Despite the claims of some sites, I can find no evidence that Hitler himself said 'Heil Hitler', and it would make no sense for him to do so as the words were more than a greeting: they indicated obedience to the leader (Hitler) and were an oath of allegiance (to Hitler). This was made clear by a senior Nazi official, Gregor Strasser, as early as 1927.

The Boothby citation by Boris Johnson is wrong. Boothby's own account is that Hitler greeted him by saying his own name (without 'Heil') and lifting his right arm, and Boothby responded in the same manner.

When Boothby said 'Heil Boothby', it was in response to Hitler's secretary saying 'Heil Hitler'; he did not say this to Hitler himself, and Hitler did not say 'Heil Hitler' to Boothby.

Albert Speer, in his memoirs, relates that Hitler would sometimes respond to Heil, mein Fuhrer from favoured members of the inner circle with Heil, Speer (or Heil, Goering / Goebbels etc.).

Details

The citation by Boris Johnson is wrong (Wikipedia's Lord Boothby has it without the full context, making it potentially misleading rather than plain wrong). Boothby was actually responding to the 'Heil Hitler' greeting from Hitler's secretary, not from Hitler himself. Rob Baker in High Buildings, Low Morals on Boothby says:

in Germany he was once greeted by Hitler’s secretary with a ‘Heil Hitler’, to which Boothby’s admirable response was ‘Heil Boothby’

Leslie Mitchell's biography Maurice Bowra: A Life clarifies further. Sir Maurice Bowra was a British academic and a friend of Boothby's from their time at Oxford University. Mitchell, citing both Bowra and Boothby himself, writes:

At some point, he [Bowra] and a party of journalists were given an audience by Hitler where they were treated to an harangue on the theme that the unity of Germany would not be political or economic, but spiritual.²⁴ It was on this occasion allegedly that Maurice replied to a salutation of ‘Heil Hitler’ with a spirited ‘Heil Bowra’. Unfortunately, there is no truth in the story, though it circulated widely and, as Maurice put it, ‘brought me nothing but credit’.²⁵ What in fact happened was more prosaic. Bob Boothby was greeted by Hitler’s secretary with a ‘Heil Hitler’, to which he responded ‘Heil Boothby’. Later, as he remembered, ‘the story was attributed to Maurice Bowra, who asked me whether I would mind if he did not contradict it. I said I would be delighted to share the honour with him, because I knew that if he had been in my place, he would have done the same.’²⁶

Footnotes (24) and (25) are citing Bowra's own Memories (London, 1966). The source cited in footnote (26) above is Lord Boothby, Recollections of a Rebel (London, 1978). Here, Boothby gives his version of what happened when he met Hitler (not the secretary) - and it's missing the word 'Heil':

It is true that when I walked across the long room to a corner in which he was sitting writing, in a brown shirt with a swastika on his arm, he waited without looking up until I had reached his side, then sprang to his feet, lifted his his right arm, and shouted ‘Hitler!’; and that I responded by clicking my heels together together, raising my right arm, and shouting back: ‘Boothby!’

From this context, it is apparent that Hitler was simply greeting Boothby and giving his name, unnecessary though it may seem. Of related interest, it was not uncommon in Jyland / Jutland (Denmark) even in the 1970s for adult men to introduce themselves to other adult males by simply saying their family name and nodding (no 'Hello' or such like, no first name).

Clearly, others (or even Boothby himself at times) have embellished the original version. For example, Israeli politician and journalist Yair Lapid's biography of his father Tommy Lapid has this:

Lord Robert Boothby…once told me the following story: “In 1931 I visited the Fuhrer in Munich. When I entered his office he jumped from his seat, stood at attention, raised his right arm in the Nazu salute and shouted, ‘Heil Hitler’ I had no choice but to stand to attention as well, raise my right arm and shout, ‘Heil Boothby’”

Who embellished this account is not in evidence; it could well have been Boothby himself for obvious comic effect.

Hitler's (various) physical responses when greeting people are well-documented (see Nazi Salute, photos and contemporary film footage). In addition to the salutes described in Wikipedia and seen in film footage, Hitler usually shook hands when greeting important or well-known people.

How he greeted them orally is less clear, but it most likely depended on the circumstances, the person or people and, quite possibly, his mood. As already stated, I can find no evidence that he used the words 'Heil Hitler' together, though this Quora post (unsourced) says otherwise (I've asked for a source so we'll see...). However, it would make no sense for Hitler to pledge loyalty to himself.

The ‘Heil Hitler’ greeting, Strasser wrote in January 1927, was not only a symbol of personal dependence on the Leader, but contained in itself the pledge of loyalty.

Source: Ian Kershaw, 'Hitler 1889-1936: Hubris' (1998)

Boothby's account is one way Hitler greeted people he hadn't previously met, but he certainly wouldn't have said his family name in this manner to children or to people he already knew.

Albert Speer's memoires Inside the Third Reich, relate that Heil Hitler was rarely used by the 40 to 50 people granted access to Hitler's 'afternoon dinner table' at the Chancellery; Guten Tag was more common. Hitler himself was fairly informal at these gatherings, shaking hands and perhaps enquiring about their well-being or sharing snippets of news. On the phone, Speer would end a conversation with Heil, mein Fuhrer', to which Hitler

sometimes replied, "Heil, Speer." This greeting was a sign of favor which he only rarely vouchsafed to Goering, Goebbels, and a few other intimates; underlying it was a note of faint irony at the mandatory, "Heil, mein Fuhrer."

There is also no evidence in Hitler's written correspondence that he used Heil Hitler, but it was used by others sometimes (see, for example, this Martin Bormann letter).

Fans of old films may have been reminded of the line Heil Myself! in the 1942 satire To be or Not to Be, directed by Ernst Lubitsch, with Jack Benny playing a Polish actor who plays Hitler. Although wholly anti-Nazi, it was considered inappropriate by many at the time despite its Jewish director and star. The film is now considered a classic. Clips with Heil Myself! can be seen here (the first 20 secs) and here (1m 39 secs, at the end of the clip). The same line was also used in the Mel Brooks-directed 1963 film The Producers.