Someone else actually asked a similar question a few months back. My answer was: there is some evidence that the Soviets attempted to infiltrate and co-opt the civil rights movement — they weren't providing support to the movement out of solidarity, but rather saw an opportunity to foment dissent and attempt to destabilise American civil/political society.

The KGB's 'Service A' was tasked with carrying out "active measures" around the world — these operations varied in nature from misinformation, to political propaganda, to assassinations. In the case of the civil rights movement, the KGB hoped to use MLK and his colleagues as a political weapon against the US. When that effort failed, they instead tried to discredit him, in the hopes of opening the door to the emergence of a more compliant, sympathetic leadership.

The below is from a book I quote regularly and at length as one of the definitive sources on the history of Soviet foreign intelligence — The Sword and the Shield by Christopher Andrew and KGB defector Vasili Mitrokhin:

King was probably the only prominent American to be the target of active measures by both the FBI and the KGB. By the mid-1960s the claims by the [Communist Party of the USA] leadership that secret Party members within King’s entourage would be able to “guide” his policies had proved to be hollow. To the Centre’s dismay, King repeatedly linked the aims of the civil rights movement not to the alleged worldwide struggle against American imperialism but to the fulfillment of the American dream and “the magnificent words of the Constitution and the Declaration of Independence.”

[...]

Having given up hope of influencing King, the Centre aimed instead at replacing him with a more radical and malleable leader. In August 1967 the Centre approved an operational plan by the deputy head of Service A, Yuri Modin, former controller of the Magnificent Five, to discredit King and his chief lieutenants by placing articles in the African press, which could then be reprinted in American newspapers, portraying King as an “Uncle Tom” who was secretly receiving government subsidies to tame the civil rights movement and prevent it threatening the Johnson administration. While leading freedom marches under the admiring glare of worldwide television, King was allegedly in close touch with the President.

On the other side, King's opponents sought to cast him as a communist sympathiser and a Soviet stooge in an effort to discredit him as a voice in the US political discourse. This was the height of the Cold War, and the height of J. Edgar Hoover's COINTELPRO operations, by which the FBI attempted to systematically undermine groups they deemed "subversive" — particularly the CPUSA, but also the civil rights and black nationalist movements, and others; especially those they deemed likely targets for infiltration by communists.

In the event, active measures against MLK and the civil rights movement largely failed in co-opting the movement, but it may have contributed to the FBI/Hoover's conviction that King was a communist fellow traveller. But the FBI's efforts to discredit him also proved ultimately unsuccessful; COINTELPRO became a national scandal after Hoover's death, and in the context of the Church Committee investigations.

(As an aside: there's a great deal of scholarship and research on COINTELPRO — you can read a huge amount of primary source material online. I'd also look at The FBI and Martin Luther King, Jr. by David Garrow and the more recent Enemies: A History of the FBI by Tim Weiner.)

Were there communists involved in the civil rights movement? Probably/almost certainly. Were they a prominent force in that movement? No, not really — certainly not as much as the FBI believed, or conspiracy theorists will continue to tell you.

Edit: new link.

I don't know much about Soviet involvement, but there was certainly domestic communist involvement in the civil rights and black liberation movements as a minority position. Most notably, this took the form of the Black Panther Party for Self-Defense. The Black Panthers were Marxist-Leninists whose radical black liberation plan was a multi-racial socialist revolution, rather than the black nationalism of other alternative black power groups. In practical day-to-day activity, the BPP was primarily concerned with community organizing and community self-defense (a concept that led to violent clashes with the police).

The Panthers were one of the biggest targets of the FBI's Counterintelligence Program (COINTELPRO), which saw the FBI infiltrate them with agent provocateurs, distribute false flag materials, pursue heavy surveillance, and carry out political assassinations.

Martin Luther King, Jr. was also a major target of COINTELPRO, in which the FBI spied on him and attempted to coerce him into abandoning his movement (or killing himself, depending on the interpretation). King was never a communist, despite FBI suspicions, though shortly before his assassination, he was organizing the "Poor People's Campaign," in which he was attempting to march a multi-racial army of poor people on Washington. King's rhetoric was also taking a more economically leftist turn, though the civil rights movement had long had ties to the labor movement and other leftists. The mainstream civil rights movement's political and economic positions can better be understood through these links, whereas the Panthers were certainly the most radical element.

As for international communism, the revolutionary government in Cuba had ties to the black community in the United States ever since Fidel Castro stayed in Harlem during his 1960 visit to the United Nations. Che Guevara, in his 1964 speech at the UN, condemned the abuse of blacks in the United States, along with a condemnation of Apartheid. Pre-revolutionary Cuba was institutionally racist in fundamental ways. After the Cuban Revolution took over in 1959, racial discrimination was declared illegal and effort was taken to bring about racial equality, though the effects were mixed and some racial problems persisted in Cuba with little state concern. Several Black Panthers fled to Cuba to escape arrest in the United States, including co-founder Huey Newton, who returned to the U.S. after a time. Assata Shakur, the only woman on the FBI's most wanted terrorists list, remains in Cuba to this day.