Traditionally, Presidents have had prior political experience and a dedicated cadre of people loyal to that person prior to the election. This has allowed them to fill positions among top staff and cabinet officials with people who are known quantities. People who have served through at least two campaigns (one for previous office and the presidential campaign). So positions like Chief of Staff, Communications Director, and Press Secretary tend to be filled by the people who did the same role in the campaign.
This was Donald Trump's first ever campaign. He didn't have dedicated people who were with him throughout the campaign. For example, he had three or four different people running his campaign. Corey Lewandoski left after an altercation with a reporter. His role had been more of a body man for Trump but grew into running the campaign.
Paul Manafort had a different title and actually started before Lewandoski left. He left after discussion of his lobbying work became a distraction (somehow his Clinton campaign equivalent, John Podesta, avoided serious scrutiny for his, very similar, lobbying work).
Manafort was replaced by two people. Kellyanne Conway became the official campaign manager, while Steve Bannon became the campaign's CEO. Neither were traditional Washington people (Conway's a bit closer than Bannon but neither is known for the bipartisan connections).
Rather than picking one of those four to be Chief of Staff, Trump picked Reince Priebus, who had closer to the right experience. Priebus was more of an establishment Republican choice. He wasn't really the kind of person that Trump would have chosen on his own, but at the time, he made Republicans feel better (and to a lesser extent Democrats). Priebus was a perfectly normal choice rather than the kind of outsider they expected. But he never really fit with the Trump loyalists.
John Kelly may fit better. He's establishment enough to get respect from Washington people but as a former general rather than a political person, he may be outsider enough to keep the loyalists happy.
Barack Obama had four Chiefs of Staff in his first term. One was considered an interim appointment. George H. W. Bush had three in his one term. So turnover there is not unknown. Trump was likely more vulnerable to it in that he had no relation with Priebus or other natural choices prior to the campaign.
At Communications Director, it's not uncommon for a new president to take some time working out who should have the position. Obama had three in his first year. If we take Sean Spicer as just a temporary placeholder, then Trump's next will be his third.
Trump is only on his second Press Secretary. Harry S Truman, Gerald Ford, and Bill Clinton also had press secretaries with short tenures. For example, Clinton's George Stephanopoulos tried (like Spicer) to be both Communication's Director and Press Secretary at the same time.
At FBI Director, Clinton also fired an FBI director (William S. Sessions). Special Prosecutor Ken Starr would eventually be appointed to investigate Clinton, an investigation that would have started under Sessions if he had been kept.
Trump's tenure has been notable not so much for the individual events but that they have all come in combination. My personal belief is that this is primarily a result of electing someone without a political background. Trump doesn't have a known pool of people competing for these positions who have already been vetted in political appointments. This leaves him vulnerable on multiple fronts. Some prospective appointments have failed vetting. Some have failed in the relationship with Trump.
If this were happening on a smaller scale at the gubernatorial level, this would hardly be news. If Trump had stronger relations with other politicians, as Obama, the Bushes, and Clinton had, then there would be more room for him to act. But Trump has just two years in politics.
This is part of his appeal. He's an outsider to Washington. But it does make it harder for him to be effective. He doesn't have the large apparatus of support that more traditional candidates would have. This is part of why his greatest successes have been judicial appointments and regulatory changes. In those areas, he's working with the Republican establishment.
