As of last year, you can read then-FBI Director J. Edgar Hoover's 1966 memo on the Soviet leadership reaction to Kennedy's assassination online. Unfortunately for present purposes, that is definitely too recent for it to have been subject to proper historical analysis and published on. But I think if we tread carefully with that memo in light of other evidence, especially the parallel situation with Cuba, we can at least get a fuller sense of the Cold War politics involved in reactions to the murder.
First of all, it's important to understand why Cuba and the USSR were concerned. Lee Harvey Oswald had spent several years in the USSR, had married a Russian woman who then immigrated to the U.S. when he returned, and--only months before the assassination--had spent a significant amount of time in Mexico applying to the Cuban and Soviet embassies for visas. (He was denied.) The dots were there to be connected by any enterprising conspiracy theorist or just a 1960s version of an idle Internet commentator.
On the surface, the Soviets reacted as a foreign power paying respects to a fallen leader ought. Russian leader Khrushchev issued solemn remarks about what a loss to the world and international relations was Kennedy's assassination. This wasn't just empty rhetoric, according to Kennedy biographer Robert Dallek--despite early disasters like the Bay of Pigs invasion, many people at the time really did see Kennedy as "a humanist arguing the case for a rational world," as well as a patriotic American, looking at things like negotiations towards nuclear de-escalation.
In another symbolic gesture, Khruschev sent his first deputy, Anastas Mikoyan, to participate in Kennedy's funeral procession. The family and the country accepted.
According to Dean Rusk's testimony to the Warren Commission, the formal U.S. government inquest into the assassination, the USSR immediately broadcast the news via Radio Moscow (here referred to as Voice of Moscow, borrowing the "Voice of America" nomenclature). The commission was actually more immediately concerned with what the Voice of America had broadcast to Russia, though!
Rusk stressed the depth and sorrow of the Russian reaction:
I think also that it is relevant that people behind the Iron Curtain, including people in the Soviet Union and including officials in the Soviet Union, seemed to be deeply affected by the death of President Kennedy. Their reactions were prompt, and I think genuine, of regret and sorrow. Mr. Khrushchev was the first to come to the Embassy to sign the book of condolences. There were tears in the streets of Moscow. Moscow Radio spent a great deal of attention to these matters.
Rusk does go further into American projection of Soviet fears (of being tapped as responsible) on a political level, or at least, what he was able to share. This is important because of interdimensional political chess--understanding how your opponents think you will react, if you are trying to avoid nuclear war, is pretty important for avoiding nasty surprises. Rusk agreed with the idea that the Soviet leadership had a genuine respect for Kennedy and would benefit from his remaining in office:
I think also that although there had been grave differences between Chairman Khrushchev and President Kennedy, I think there were evidences of a certain mutual respect that had developed over some of the experiences, both good and bad, through which these two men had lived. So that it would be an act of rashness and madness for Soviet leaders to undertake such an action as an active policy. Because everything would have been put in jeopardy or at stake in connection with such an act. It has not been our impression that madness has characterized the actions of the Soviet leadership in recent years.
Rusk also cast doubt on the possibility that a radical breakaway wing of the Soviet Communist Party was responsible, pointing out that such a group would make a lot more progress on their goals by, you know, assassinating other Soviets.
These points all dovetail with information presented in the Hoover memo:
But also, Rusk's implicit point tracks with one of the major thrusts of Hoover's memo. The overall arc of his testimony is to disavow Soviet involvement in Kennedy's murder. The real question he's answering, through all of it, is whether they were involved, and he says no. That's the American concern, that's the Soviet concern.
That's one of the reasons the Warren Commission works here as evidence. Obviously there is A TON not being said, and probably a good share of truth bendings and outright lies. But the public perception that the testimony is trying to create is important evidence in and of itself. (Especially if you consider that the KGB probably had sources telling them about internal classified dealings as well).
Cuba's involvement was perhaps a hotter issue. It was heightened because the U.S. and Cuba did not really have established diplomatic ties at this point in time. The small matter of the U.S. assassination plots against Castro headlined by, oh, the U.S. Attorney General, that would be the brother of the murdered president, might have had something to do with that along with heightening the tension...Cuba, says Rusk, was an unknown quantity.
And actually, that past bloodstain was the reason for a lot of U.S. concern. Both Robert F. Kennedy and Lyndon Johnson worried that Castro or Castro supporters were part of a conspiracy behind Oswald. According to Schlesinger and Holland, Bobby Kennedy in particular tortured himself endlessly over the assassination, believing his role as AG in the anti-Castro plots had gotten his brother killed in revenge. It probably didn't help that an interview with Castro published in numerous U.S. newspapers just that September quoted him on the assassination plots to the effect of:
CASTRO: Ware prepared to fight them and answer in kind. United States leaders should think that if they are aiding terrorist plans to eliminate Cuban leaders they themselves will not be safe.
U.S. official response, obviously, followed Warren Commission testimony that Cuba likewise had had no involvement.
Overall, though, the official response was for the Soviets to declare their deep sadness, and America to insist on a lack of Soviet conspiracy. Those two points, in addition to being honest, definitely give an undercurrent of worry on both sides that if the Soviets were to be found at blame (justly or not), the result could be devastating to everyone.