How bad was the relationship between Richard Nixon and Dwight Eisenhower?

They really didn't have a bad relationship. Eisenhower had no objection to his grandson David (for whom the Camp David Presidential Retreat is named) from marrying Nixon's daughter.

Nixon would not have challenged Eisenhower in the 1956 primary. No sitting VP would EVER challenge a sitting President who was running for another term. Not going to happen.

As for replacing Nixon in 1956. Yes it was considered. Why? Eisenhower thought he would be doing Nixon a favor. Back then VP nominees were selected to balance the ticket. Eisenhower was a Moderate with occasional Liberal leanings from Pennsylvania (born in Kansas but living in Pa) on the east coast while Nixon was a Moderate with occasional Conservative leanings from California on the west coast. No consideration was made whether the VP was ready to be President should something happened. Also, once in office, the VPs were generally ignored and not apart of the policy-making process. They generally were sent around the world on tours or to funerals of ex-leaders. In 1960 when Nixon was running for Pres, Eisenhower was asked what Nixon did regarding a policy decision & Eisenhower responded that if the reporters gave him a few days he might think of one. Nixon, like his predecessors and a few of his successors weren't allowed to be involved except for the National Security Council & only because a 1947 law requires it.

In 1956 Nixon only had a little experience--4 years in House and 2 years in Senate when he became VP in 52. As I said Eisenhower, a political novice who announced he was a Republican in 1952, thought he would do Nixon a favor by getting Nixon some Executive experience before Nixon did his 1960 campaign. To do this Nixon would be a Cabinet Officer running 1 of the Departments. Nixon, the man with limited political experience had to tell the political amateur Eisenhower how destructive this DEMOTION would be to Nixon's political career. Eisenhower understood & let Nixon stay on as VP for the 56 campaign.

Yes Eisenhower was late to endorse him in 60 but that was in part due to Nixon being challenged for the nomination. Nelson Rockefeller, representing the Liberal Wing of the Republicans, put out feelers but then chose not to run. Barry Goldwater, representing the Conservative wing, did run but lost to Nixon.

There was one major thing about Nixon that Eisenhower didn't like....Nixon's support for Joe McCarthy. Sen. Joe McCarthy was running around the country calling major Democrats communists and traitors to America. One of the people McCarthy called a traitor was George C. Marshall, Secretary of State under Harry Truman, but a close friend of Eisenhower. Nixon did the same thing as McCarthy which is how Nixon won the 1950 US Senate campaign. Eisenhower hated McCarthy calling the man who was the Army Chief of Staff in WW2 and who was a mentor to Eisenhower a traitor. Both Eisenhower and WW2 Army General Omar Bradley felt that they wouldn't have gotten as far as they did in the Army if it wasn't for Marshall. For McCarthy to call a 5-star General of the Army who helped planned the US' WW2 a traitor to America burned Eisenhower up and here was a VP---Nixon--who was friends with McCarthy, approved of what McCarthy was doing & who did it himself. During the 1952 election, Eisenhower had a paragraph in a speech to be given in McCarthy's home state of Wisconsin denouncing McCarthy & praising Marshall but had to be talked out of it of saying the paragraph by the Wisconsin Governor & Eisenhower's Campaign staff due to concerns that it could cost Eisenhower Wisconsin & possibly the election.

Michael gave a very thorough and clear answer, I would add one thing regarding their relationship. It's known that Eisenhower had a hidden recording system to secretly record conversations with people but more often than not he forgot to turn it on. There was one person Eisenhower never forgot to record his conversations with and that was Nixon, he never really trusted his VP.