The reaction in the USSR was much the same as the rest of the world, shocked and saddened. It was late evening in Moscow when news of JFK's death broke, so many people found out the next day. Khrushchev cut a trip to the Ukraine short and hurried back to Moscow. Soviet television showed pictures of Kennedy with somber music playing between news stories.

According to western press reports, the average Soviet citizen displayed as much shock and sympathy as anyone else. They had been impressed by Kennedy's youth and charisma, and they were concerned about the change in leadership. This is a UPI story that ran in several US newspapers about the Soviet reaction: http://www.upi.com/Archives/1963/11/23/Khrushchev-calls-Kennedy-death-a-heavy-blow/3503214243588/

Soviet press, like most of the rest of the world, initially assumed that Kennedy's assassination was the result of a right-wing plot. However, when it was learned that Oswald was actually a left-wing activist and had lived for a time in the USSR, that information was kept from the Soviet public for a long time.

The next day Khrushchev and his wife visited the US embassy in Moscow and signed the condolence book (considered the highest sign of respect). Khrushchev had tears in his eyes and his wife left a personal note with the ambassador to pass along to Lyndon Johnson. Jackie Kennedy wrote a very warm and appreciative note to the Soviet leader a week later, which you can read here: https://history.state.gov/historicaldocuments/frus1961-63v06/d120

Soviet officials quickly realized, through the KGB, that Oswald was in no way connected to any part of their government. They saw him as a mentally troubled person and he would not have been a candidate for recruitment. They communicated as much to the Americans. When the Warren Report came out, it showed the Americans had reached the same conclusion.