This is not exactly what you are asking for, but perhaps the reaction of Czechoslovakia, a member of the Eastern Bloc at the time, might be of some interest.

On November 23, Czechoslovak president Antonín Novotný sent his condolences and sympathy to Lyndon B. Johnson, and expressed his belief that the assassination was the result of a criminal conspiracy.

On the same day, Kennedy's death made it to the front pages of local newspapers, most notably Rudé právo, the official communist party paper and the flagship media outlet in socialist Czechoslovakia. This is what the title page read: "President Kennedy assassinated in Texas - perpetrators unclear - vicepresident L. Johnson to take office - far-right behind the crime?"

The early coverage voiced similar concerns as the president: that Kennedy was not killed by Lee Harvey Oswald acting independently, but that there are "conflicting reports" as to who might be behind the killing. Besides the far-right extremists, "rumours of FBI or CIA involvement" were also mentioned.

In addition to this speculation, the very first reports of the assassination also recalled the Adlai Stevenson incident, noting that he was attacked by "far-right groups and racists" in Dallas recently and that he also was a Democrat, in fact a representative of the more liberal wing of his party.

That far-right extremists were behind the assassination, that there were more shooters or at least an organized group of co-conspirators, and downplaying of Oswald's communism were standard assumptions at the time and throughout much of the sixties. The act of killing Kennedy was also shown as proof of the rotten state of democracy in the United States – "a country where a president is killed just because of his opinions". Furthermore, doubts were cast on the ability of the American government to fully investigate not only the assassination of JFK himself, but the killing of Oswald by Jack Ruby.

The weeks following Kennedy's death also brought a change in his image in Czechoslovak press. Here is a lovely study of 260 Kennedy-related articles in three major newspapers, Rudé právo, Lidová demokracie and Mladá fronta, from September to December 1963. It is in Czech, though the newspaper clippings at the end of the document and the tables that follow might be of some interest and easy to make sense of using an online translator.

Basically, before Kennedy's assassination, the articles were "neutral" or "rather negative" in their reporting. Immediately after his death (November 23 - 26), they – perhaps unsurprisingly – tended to be "rather positive" or even "(clearly) positive". But even pieces from November 27 onwards kept the more positive evaluation, albeit not as strong anymore. Local press also mentioned that the reactions in the Soviet Union were quite similar and just as sympathetic, but I would rather leave that to others.

However, Kennedy was actually praised by Czechoslovak newspapers on occasion even before his assassination. The press labeled him a good diplomat, a realist, and lauded his attempts at improving US-Soviet relations. But it should come as no surprise that American involvement in South America and Asia as well as perceived social injustice and racism were perhaps the most notable topics which earned him and the United States much criticism in Czechoslovak newspapers.

The negative view of US foreign policy remained even after his death, but it took a back seat to showing the late JFK in a quite positive light and expressing regret at his passing.

A related question: was there any real belief in the US that the Soviets may have been behind it?

Since the user with the wonderful name /u/CCCVCCCC pointed out the situation in Czechoslovakia, I got curios about the GDR. Here's what I found:

On November 23, 1963 the communist party paper Neues Deutschland reported about the incident as a 'work of murder by right wing extremists' which had happened in "the darkest center of reactionary" forces (a label given to all anti-socialist movements back in those days). It also speaks about condolences sent by then head of state Walter Ulbricht.

The papers of the following days keep the story on the front page and also cover the reaction in the Soviet Union claiming "deep sympathy of the Soviet Union being expressed in telegrams from Khrushchev and Brezhnev to President Johnson and Mrs. Kennedy." Unfortunately the archive is behind a paywall. The headlines keep claiming that "the world accuses ultras of the murder".

A comment on November 24th states, "the world is under the impression of the terrible crime, to which the President of the United States has fallen victim. Greatest abhorrence of the murderers and sincere sympathy with the American people [are expressed...]". On Nov. 25th the paper already claims about the murder of L.H. Oswald, that "political observers expressed the view that the perpetrators in this assassination are to be sought in the same right-wing extremist circles, which also had President John F. Kennedy killed."

As to the OP's question:

From an FBI document that was de-classified and released in 1997:

"A source who has furnished reliable information in the past and who was in Russia on the date of the assassination of the late President John F. Kennedy advised on December 4, 1963, that the news of the assassination of President Kennedy was flashed to the Soviet people almost immediately after its occurrence. It was greeted by great shock and consternation and church bells were tolled in the memory of President Kennedy.

According to our source, officials of the Communist Party of the Soviet Union believed there was some well-organized conspiracy on the part of the "ultraright" in the United States to effect a "coup." They seemed convinced that the assassination was not the deed of one man, but that it rose out of a carefully planned campaign in which several people played a part. They felt those elements interested in utilizing the assassination and playing on anticommunist sentiments in the United States would then utilize this act to stop negotiations with the Soviet Union, attack Cuba and thereafter spread the war. As a result of these feelings, the Soviet Union immediately went into a state of national alert.

Our source further stated that Soviet officials were fearful that without leadership, some irresponsible general in the United States might launch a missile at the Soviet Union. It was the further opinion of the Soviet officials that only maniacs would think that the "left" forces in the United States, as represented by the Communist Party, USA, would assassinate President Kennedy, especially in view of the abuse the Communist Party, USA, has taken from the "ultraleft" as a result of its support of peaceful coexistence and disarmament policies of the Kennedy administration."

Link: http://www.indiana.edu/~oah/nl/98feb/jfk.html

Not a Soviet reaction, but I found an article in a local Slovene (then Yugoslavia) newspaper, where page 8 of November 27th issue writes a few lines about the assassination. (Screenshot) It writes:

Shots in Dallas

United States of America and the rest of the world was shocked by unexpected and ghastly news last week: the president John Fitzgerald Kennedy was a victim of assassination. He is lying in Parkland Hospital in Dallas, his status being critical. Twenty minutes later, another, even more shocking news came: Kennedy is dead.

We write this on Monday and it has been three days since the event. Right now, Kennedy's funeral is being arranged in Washington. He is to be buried on Arlington cemetery, while the funeral will be attended by highest dignitaries and famous people from all over the world.

Kennedy's sudden death is a grave personal and political tragedy by itself, since the late president was a respected man both in personal life and in politics. He strived for a progressive legislation (battle against racial segregation, social legislation) and for better relationship between East and West. But the circumstances surrounding his death spell treachery, confusion and uncertainty. Especially if events leading to assassination are taken into account.

Police have arrested 24-year-old Lee Harvey Oswald in not quite clear circumstances. He is said to had been in Moscow and that he is a communist. He reportedly killed the president from the fifth floor of a building which was supposed to be emptied by the police. Despite intense interrogations that lasted for two days, Oswald denied his involvement. He was saying he was guilty up until he himself was shot by a 50-year-old Jack Ruby, the owner of a nightclub Carousel in Dallas. Oswald died in the same hospital as the president.

The assassination of the alleged killer complicated things even more. The local police stated it has strong evidence that Oswald was the killer, but they have not presented them to the public. The moment the evidence was supposed to be presented, the nightclub owner showed up conveniently and shot the only suspect.

The biggest question now is how did Jack Ruby get the access to kill Oswald, despite the extremely tight security around him. Police say they knew him and that was the reason they didn't pay attention to it. But still: Jack Ruby had no business around Oswald.

Soon after Oswald's death the American and the World public asked themselves: What now? Was Oswald the real killer? If he was, we will never know, why he insisted that he was innocent up until his death. Haste with which the Dallas police is trying to wrap up things is suspicious as well. Right after Oswald was killed, they said the case is closed and continuing the investigation makes no sense. But what about strong evidence?

Even if they could prove that Oswald was guilty, there are still many unanswered questions. But the Dallas police did not prove anything. What is going on in Dallas, in this nest of reactionism and fascism? What is the real background of the Dallas tragedy? Did Oswald really kill Kennedy? And if he didn't, who did?

Those questions have not been answered, but America should insist on investigating it. The Dallas murder was no ordinary murder, but an assassination of a certain kind of politics. After all has been said and done, the picture should be clearer. But contrarily, the picture is even murkier. And that does not mean anything good.

(Edit: added the screenshot and a few spellchecks)