Is there any consensus among modern day historians of how Hitler was killed or if he was killed, and is there any significant proof that he did die so as to disprove the conspiracy theorists?
Because the death of Hitler was reported by the Soviets, this is generally the root cause of conspiracy theories.
The general consensus among historians is that Hitler committed suicide in his Führerbunker in Berlin, by gunshot on 30 April 1945, however, controversy will remain.
His body was found by a soviet counter-intelligence operations group called SMERSH, this in itself is a cause of distrust for the first hand account.
This trust problem is furthered by disinformation from the Soviets, for example when President Truman asked Stalin at the Potsdam conference in August of 1945 if Hitler was dead, Stalin simply replied 'No'.
Over time, the story surrounding Hitler's fate was variously presented by the Soviets in line with their political motivations, in the years immediately following 1945, it maintained Hitler was not dead but had fled and was being shielded by former western allies.
At the time this worked, and it was stated at the Nuremberg trials that "No one can say he is dead." (Thomas J. Dodd, chief of the U.S. trial counsel.)
The final account from the Soviets was that Hitler and Eva Braun, were buried at a SMERSH facility in Magdeburg, the exact location was kept a secret to avoid the site becoming a shrine for Neo-Nazis. It was also reported that the site, and the remains were destroyed.
This layering of misinformation, and the justifiable need for secrecy has left the ultimate fate of Hitler a mystery, which will never be truly solved.
There is some controversy of what happened to Hitler's remains, owing to the general disarray of war, but there is no real controversy with regards to his death.
Hitler did in his personal will and testament say that he had chosen death.
I myself and my wife — in order to escape the disgrace of deposition or capitulation — choose death. It is our wish to be burnt immediately on the spot where I have carried out the greatest part of my daily work in the course of a twelve years' service to my people.
This is consistent with the eyewitness accounts of those who survived, like Hitler's secretary, Traudl Junge, and Rochus Misch. As such there is little doubt about the general events, although the exact details may be unclear.
Update: It has recently come to light that a Swedish SS-officer was in the bunker, witnessed Hitler's wedding and saw Hitler's burnt body. He recounted this for the priest at the Swedish delegation in Berlin, who gave this information to the Swedish military, but did not confirm who the officer was until after that officers death. This description of the events also is consistent with other eyewitness accounts, and as this has been classified in the Swedish archives since 1945 it is a witness account that is independent from the other sources, significantly strengthening them.