Based on sources such as Chicago Tribune, Encyclopedia Britannica, Slate Magazine and many other books, it is said that JFK's father played a significant part in making him president. In the book, "If It's Not Close They Can't Cheat," by Hugh Hewitt (page 60-61), it is claimed that JFK's father "helped" him illegally become president. What is some of the proof for and against this?
As per your original question:
The Wikipedia section does not claim that Joseph P Kennedy bought John F Kennedy's victory. Such claims would be rather outlandish as it would mean that he was the only or the major contributor to the primary election campaign. As JFK would not have been able to win the democratic primary election unless he had widespread support within the Democratic party, it is unlikely that he couldn't find campaign funds from many other sources.
As per your updated question:
There was several investigations and recounts in this election. The only of them that changed the outcome actually changed the 3 electoral votes of Hawaii from Nixon to Kennedy.
Claims like these often show up in countries where the politics is highly polarized into blocks, as people are unable to accept defeat in something that they feel very strongly. They feel cheated after having put down a lot of emotions (and often hard work), and they decide that they feel cheated because they really were cheated.
The Republican party gave up their attempts to change the outcome of the election once JFK was in office, but as you notice, some republicans still haven't given up.
About Joe Kennedy's influence on making JFK president:
Joseph P Kennedy was involved in politics, and knew a lot about politics, and helped out his son in his political career in many ways. I don't see how this is strange or controversial in any way. And it doesn't mean he paid for the presidency.
There were indeed some serious issues surrounding Kennedy's election in 1960, one of the closest elections in History.
However, the controversies at the time involved contentions regarding voter fraud, principally in Texas, the home state of Lyndon Johnson, Kennedy's running mate, and in Illinois, particularly certain areas of Cook County (Chicago area), which was controlled by the very powerful Democratic political machine led by Mayor Richard J. Daley of Chicago. Having lived through those times, I remember all of this quite well. The radio and the papers at the time were talking constantly about the problems in Chicago that might have an impact on the election's outcome. See United States presidential election, 1960 -Controversies for verification of my primary source. :-)
As for Hewitt's idea that JFK's father was very important in making his son President through elicit means, has Hewitt unearthed some unknown and credible historical evidence supporting his contention? That there were problems with the election of 1960 is established - but 'business as usual' in American politics: Partisan warfare, corrupt political machines, potential voter fraud, etc - issues that have surfaced in so many elections throughout American history - seem to be more than sufficient to explain the 1960 election without resorting to novel and weakly substantiated theories.