To answer this, let's begin with the fact that Kennedy and Nixon both started their political careers at the same time - both were freshmen congressmen in 1947. That year, the two debated in McKeesport, Pennsylvania. According to biographer Chris Matthews, the two ate at a local grill, talked sports, and took the train back to Washington together. The two had some things in common - they both served in World War II, they both were newcomers to the political world (despite JFK's father's influence), and both shared a common goal - to become President, which would affect their friendship years later.

To say the very least, Nixon and Kennedy respected each other before the events of the 1960 Presidential election. Kennedy's father, Joseph Sr., even had his son deliver a $1,000 contribution on his behalf to Nixon's 1950 Senate campaign due to his anti-communist stance. When the 1960 election came around, JFK also personally asked Nixon to keep the donation quiet, and Nixon agreed.

In 1954, Jackie Kennedy stated that "there's no one my husband admires more" than Nixon; that same year, amid rumors that JFK was not doing well during his back surgery, Nixon broke down and said "Poor brave Jack is going to die. Oh, God, don't let him die."

After the election, however, Nixon took a different stance. Because of the close vote tally, speculation persisted that Kennedy stuffed the ballot boxes in Illinois and Texas, two states that would have given Nixon the presidency had he won them. Alexander Haig, a close friend of Nixon, stated that despite the former Vice President conceding gracefully he believed the election was rigged until his death - a theme Nixon also mentioned in an autobiography. This also affected him in his 1968 and 1972 campaigns, for there had been a possibility of him facing a Kennedy there as well - Robert for the former, and Edward for the latter.

Once JFK was in office, Nixon began attacking him on his handling of the Cuban Missile Crisis (at least, as suggested by Matthews - I couldn't find anything), and JFK fully expected Nixon to get involved in the 1964 election. This, however, never came to fruition due to Kennedy's assassination.

In conclusion, JFK saw Richard Nixon as somewhat of a friend, but also a rival during their time in Congress and vying for the Presidency. Nixon, at least for some time, felt the same way.