Explain the connection between the cold war and human rights including civil rights

  As far as I'm concerned, the relationship between the two begins with Eisenhower. At the Tehran Conference, Stalin invited Ike to tour Russia after the war. Ike did and took his son John with him. Stalin had villages erected along Ike's "parade route" and then filled them with "happy" peasants. 

As President, Ike called upon Congress to pass Civil Rights legislation and then pushed forth policies of desegregation of both the military and the public schools. His call for legislation was met with fierce resistance (by the democrats) as was the desegregation of the public schools. For that, he was forced to call out the National Guard. 

Ike also began flying secret spy missions over the Soviet Union, using older B-29s re-equipped with all sorts of radio and photography equipment. As technology quickly progressed, he switched to using the Lockheed U-2 spy plane which could fly so much higher than anything the Russians had. And then they actually got lucky and shot one down. The CIA had assured Ike that it would never happen and that even if it did, both the pilot and aircraft would burn up re-entering the earth's atmosphere. Neither happened and the Russians were able to present wreckage from the downed aircraft as well as a battered but living Capt. Gary Francis Powers USAF. 
Ike was a Republican. 

Under the Kennedy administration Civil Rights sort of floundered. Rev. Martin Luther King, Jr. pushed the issue and Kennedy tried (unsuccessfully) to duck out of it. The south and inner cities were burning with racial tension. Kennedy placed nuclear missiles in Turkey and then got called on it when Russia began deploying nuclear missiles in Cuba. Kennedy was forced to quietly recall the missiles from Turkey and in return, the Russians would pull theirs out of Cuba. Kennedy then tried invading Cuba with some 2,510 men and four armed aircraft. All died because after cancelling the support for the mission, he then notified the CIA that the mission was still on. Nice guy eh? Kennedy then involved the U.S. in Vietnam. Kennedy was a Democrat. 

Next came Johnson who tried to fashion himself to be the next great FDR. He pushed for Civil Rights legislation and again his fellow Democrats fought it by filibustering. Still the south and inner cities were on fire with racial tension. Johnson increased the military role in Vietnam bringing it to an all out war. He also got us more deeply involved in CIA operations in Central America. 

And on and on it goes until Reagan bankrupted the Soviet Union.