The immediate reaction from the Soviet government was sympathy and cooperation. The Wilson Center Digital Archive has some great primary documents concerning the Soviets' immediate response here, including the letters approved and sent by Soviet Premier Nikita Khrushchev to President Lyndon Johnson and JFK's widow Jackie Kennedy, offering his sympathies and condolences. Khrushchev's successor, then Chairman of the Presidium of the Supreme Soviet of the USSR, Leonid Brezhnev also sent a message to LBJ offering condolences, and Khrushchev's wife Nina also sent a personal message to Jackie Kennedy.

Khrushchev's letter stated that he viewed the assassination of the late President as "heinous" and and that it had "evoke[d] the indignation of the Soviet people towards the perpetrators of this vile crime."

These documents also reveal that the Central Committee held a meeting on the day after the assassination to discuss their response to the Kennedy assassination, the summary of which was titled "On Measures [to be Taken] in Connection with the Murder of U.S. President J. F. Kennedy" and how to deal with the funeral. They appear to have concluded that their steps should be to send telegrams offering condolences, such as the ones sent to LBJ and Jackie Kennedy, as well as telegrams instructing Anastas Mikoyan and other Soviet diplomats to Washington to attend JFK's funeral.

Another document, which you can view here, provides a telegram sent from the Soviet Ambassador to the United States Anatoly Dobrynin to Moscow requesting advice concerning the news that Lee Harvey Oswald, who had communist connections, had been arrested and had been reported to have spent time in the USSR. Dobrynin states that after checking, it appeared the Oswald had indeed spent time in Minsk and had a Russian wife. Dobrynin notes that the media were focusing "more and more" on Oswalds left-wing sentiments and connections. Dobrynin suggested that the Soviets may wish to cooperate with the US government in providing Oswald's correspondence with his Russian wife.

In a telegram to Moscow from Andrey Gromyko, Gromyko stated that "slanderous fabrications" were being made by the US media with regard to Oswald's connections to the Soviet Union and Cuba. Gromyko suggested that if the US government requested information on Oswald's stay in the USSR, the embassy in Washington could provide them with a report. The reply from the Central Committee in Moscow approved this suggestion.

In all correspondence between Moscow and Washington, the Soviets were sympathetic and condolent. Soviet diplomats in Washington were worried that Oswald's connections with Cuba and the USSR would cause the US to believe the Soviets were involved in JFK's assassination, and the correspondence between these diplomats and Moscow suggest that the Soviets were keen to distance themselves from the assassination, and wanted to cooperate so as to ease tensions inevitably caused by Lee Harvey Oswald's communist connections.