Ah... Nixon and JFK. So opposite in many ways, yet so symmetric in others. That relationship never ceases to capture to the imagination.

Although always the political partisan, Nixon staunchly supported Kennedy during the Cuban Missile Crisis(though he would be heavily critical of Kennedy after the fact), and unlike the Goldwaterite wing of the party, supported much of the "in memory of Kennedy" powered legislation passed by Johnson in 1964, notably the Civil Rights Act. That being said, I definitely think he thought that JFK was overrated. It's hard to avoid coming to the conclusion that Nixon thought he could do far better than Kennedy on foreign policy. This wasn't limited to Nixon-LBJ also clearly thought the same, but with Congressional legislation.

It is hard to avoid coming to the conclusion that both were right to a certain degree. Though Kennedy set the initial path for both the Civil Rights movement and arms control with the Soviet Union, Johnson's Civil Rights/Great Society legislative bonanza and Nixon's overall reshaping of geopolitical dynamics from 1970-1974 went far beyond anything Kennedy had a prayer of doing, had he lived. Part of this was the time period, but part of this was indeed ability. Both men stacked up unholy electoral landslides as a result that Kennedy in 1964 likely wouldn't have come close to approaching. However, both men also had personality demons that would sabotage their rule. I'm not sure what would have happened if JFK had lived-certainly not the Boomer fantasy utopia-but I doubt he would have followed their path. (LBJ and RFK's absolute loathing for each other couldn't have been helped by that.) Though Nixon was never crass enough to be explicit about it, there's no doubt he thought that Kennedy won by less than fair means-his decision to not challenge the 1960 results was far more complicated than portrayed in hindsight, and the GOP did a fair amount of ballot stuffing themselves in downstate Illinois. People around Nixon, such as his wife, were far more explicit about it.

Nixon's anger was not limited to the electoral results-he was convinced that the media and the "Best and Brightest" largely supported Kennedy (this one, to be fair, was not without reason-hard to believe that the not so subtle pro-Kennedy tilt among the nation's leading intellectual circles and East Coast media didn't make 100,000 votes of difference), that the CIA intentionally leaked to Kennedy the plans for the Bay of Pigs, hence Kennedy's ability to outflank him on Cuba in the debates (which led to his pathological distrust of the CIA), and that the Fed refused to stimulate the economy for him during the 1960 election (which led to some very poor domestic economic decisions in the summer of 1971 for his reelection). Overall, while Nixon's character was always gray, there's no doubt that 1960 definitely deeply darkened his persona, made him more bitter and paranoid.

Subsequent events didn't help. Nixon was also extremely angry at the politically motived IRS audits launched by RFK in 1961 against his family, and the illegal wiretaps ordered by RFK were something he pointed to when justifying such measures to his staff. The reason for these actions is because the Kennedys viewed Nixon as a political threat (the dirty secret here being that he wasn't "supposed" to convince near half the country that he was superior to Jack, in 1960. Jokes aside, they were clearly disturbed at how close the election was. To be fair, though, Nixon gave as good as he got in campaign rhetoric, as always.) until his implosion against Pat Brown in 1962, and treated him accordingly. The Kennedy family was friendly to all, unless you were a threat to them. I don't want to overstate this too deeply: Nixon and Kennedy were polite and civil in their post-1960 face to face meetings, if not in private about each other. Their last phone conversation in Europe in 1963, after Nixon had left politics, was actually somewhat friendly. But the relationship from the 40s had irreparably been broken.

With that being said, however, Nixon was horrified and shocked by the assassination. He had been in Dallas, ironically, just before it, and only learned of it on his way home in New York. Here is how he described it:

Arriving in New York, I hailed a cab home. We drove through Queens toward the 59th Street Bridge, and as we stopped at a traffic light, a man rushed over from the curb and started talking to the driver. I heard him say, “Do you have a radio in your cab? I just heard that Kennedy was shot." We had no radio, and as we continued into Manhattan a hundred thoughts rushed through my mind. The man could have been crazy or a macabre prankster. He could have been mistaken about what he had heard; or perhaps a gunman might have shot at Kennedy but missed or only wounded him. I refused to believe that he could have been killed. As the cab drew up in front of my building, the doorman ran out. Tears were streaming down his cheeks. "Oh, Mr. Nixon, have you heard, sir?” he asked. “It’s just terrible. They’ve killed President Kennedy.”

That night, feeling according to himself as low as he did when his brother died of tuberculosis as a youth, he would later write a letter to Jackie Kennedy pledging to be of any use he could. Jackie's response was interesting: she might have perceived of the political chance that Nixon could have in the next decade before he did.

Nixon would be haunted by the ghosts of the Kennedys during his Presidency, later on. I would recommend Larry Sabato's work for more on that. Nixon took a mostly ambiguous relationship with Kennedy, post-mortem, except for two things, both bad: Vietnam and Watergate. Nixon, obsessed with his press ratings, constantly pointed out the supposed asymmetry between their coverage, in spite of the fact that his approval ratings during 1969 were consistently higher than Kennedy's in 1961. Like Johnson, he eventually found out that he "couldn't compete with a saintly ghost", no matter how much more he achieved. There was a very eerie symmetry from beginning to end, even during Watergate: Massachusetts was the only state to go for McGovern, and EMK and Ethel Kennedy were there when Archie Cox was sworn in as Special Prosecutor.