This is no doubt a politically charged question. The most simple answer is that the laws and constitution of the the United States say that it is the public's responsibility to deem a candidate qualified or unqualified. This has not been decided yet, but if the public were to elect Donald Trump, this in itself would deem him qualified from a certain perspective.
That said, conventional wisdom states that a candidate should have experience in a related public office before seeking the presidency. Most candidates have been a senator, governor, or other official before seeking the highest office. This helps the candidate to learn the inner workings of government, make connections, and arguably become more effective as a leader. From this perspective Donald Trump is lacking in his qualifications. He does not have experience in public office, unlike many of his opponents.
Trump frequently argues that the country is currently in such a bad state that his lack of experience is actually a strength, since he is not caught up in the insider game of politics. This is a common and old argument going back for many years in the American political tradition. Certain voters always find so-called outsiders to be appealing. From the perspective of these people, it would seem his lack of political history actually makes him more qualified.
Additionally, many of Trump's supporters would argue that his success as a business man means he will be an effective leader. This argument rests on the assumption that public leadership and private leadership are related in skills and qualifications. His supporters would say that only an outsider with business experience can fix a broken system. His critics would point out that while he has done many things in his career, he would undoubtably be the least politically experienced person ever to hold the office.
Electing Donald Trump would certainly go against conventional political wisdom, but his right to lead is ultimately a decision for the American public. In a sense the only legal requirement is that the president be a natural-born citizen of the proper age.