It is true that the KGB did help fuel the early kennedy conspiracy theories, in fact, the first book on the JFK assassination published in the US that alleges a conspiracy, titled Oswald: Assassin or Fall Guy?, was written by Joachim Joesten, a German communist and published in New York by a KGB agent named Carlo Aldo Marzani.
As for your question you might be interested in Lt. Gen. Ion Mihai Pacepa. Pacepa was the former head of the Romanian secret police under Ceauşescu and the highest ranking defector during the Cold War. He witnessed and sometimes carried out the USSR's clandestine operations, especially what he called "disinformatzya." He Alleges that Oswald was a KGB agent gone rouge and was part of Khrushchev's assassination campaign that targeted world leaders (potential targets included Mao, the Shah and others). What you have to remember is KGB archives, especially those dealing with clandestine operations, have not been released to the public. We only know that the KGB was part of spreading the conspiracy theories regarding JFK because a small number of files were smuggled out of the USSR in 1992, what we now know as the Mitrokhin Archive. The vast majority of the KGB files are still sealed away in Russia.
Pacepa wrote two books the subject: Programmed to Kill in 2007 and Disinformation in 2014, I have not read Programmed to Kill but I have read Disinformation and I think that it is an excellent narrative on Soviet disinformation during the Cold War. I'm not saying I totally believe him regarding Oswald, but in Disinformation his version of events is pretty convincing. Still, I would HIGHLY recommend Disinformation to anybody looking to know more about the Cold War.
Sources:
The Sword and the Shield: The Mitrokhin Archive and the Secret History of the KGB by Christopher Andrew and Vasili Mitrokhin
Disinformation: Former Spy Chief Reveals Secret Strategies for Undermining Freedom, Attacking Religion, and Promoting Terrorism by Ion Mihai Pacepa and Ronald J. Rychlak
There have been two official government investigations into JFK's death--the Warren Commission, headed by Chief Justice Earl Warren, conducted in the immediate aftermath of the assassination and completed in 1964, and the investigation of the House Select Committee on Assassinations, which concluded its work on 1978.
Both reached the conclusion that Lee Harvey Oswald was responsible for assassinating the president. The WC concluded that he did so acting alone, while the HSCA concluded (based on sketchy evidence) that he acted with one co-conspirator who fired one shot that missed.
The majority of the evidence supports the conclusion that Oswald acted alone. Much of the fodder for conspiracy theories stem from two main sources.
First, the case against Oswald is largely circumstantial (meaning no one actually witnessed him fire the shots). That said, the circumstantial case is fairly strong and include the facts that he ordered the murder weapon by mail, worked at the building the shots were fired from, left a palm print on the murder weapon, may have murdered a police officer in the hours immediately after the assassination, and attempted to murder another police officer while he was being arrested. Those things, combined with his background as a troubled, depressed individual, leads to the likely conclusion that he was indeed the sole assassin.
Second, the investigation after the shooting was handled in a very poor, haphazard way leading to a lot of seemingly contradictory testimony and physical evidence that gives wiggle room to tons of alternative theories.
I became extremely interested in the JFK assassination a while back, and spent hours reviewing the primary source material, which is what I recommend for anyone seriously interested in learning about this topic and is the basis for my statements above. I began my research believing in some sort of conspiracy theory, and by the time I was finished concluded that I was probably wrong, and that the most likely explanation was that Oswald acted alone. There are some pieces of physical evidence that are difficult to explain in the context of a lone assassin theory which leave a little crack in the door for an alternative explanation, but it is just a small crack.
I always enjoy this topic and would love to give more details about any particular aspect of anyone is interested.
http://spartacus-educational.com/JFKbowers.htm