The Sword and the Shield: The Mitrokhin Archive and the Secret History of the KGB mentions several times the USSR either created or promoted conspiratorial thinking, but one of the take-home messages of the book is not just that the KGB liked to cynically promote conspiracies in other countries, but that the KGB and more broadly the political apparatus of the Soviet Union were prone to conspiratorial thinking, siege mentalities and confirmation bias.

Particularly in the Stalin era, but persisting well beyond it, the people interpreting and presenting intelligence would frequently frame it in ways that sought to confirm the underlying beliefs of their superiors. Intelligence coming out of Britain in the interwar period told the Soviet Union that Britain did not consider the USSR a high priority for intelligence gathering, and did not have any large scale operations taking place there. This was considered to be essentially impossible, as Stalin and many of his subordinates believed that the United Kingdom was the linchpin of a vast conspiracy to overthrow the Soviet Union's government and destroy communism. Doubt was cast on the validity of the intelligence coming out of these sources because they did not conform to these "obvious truths". A later example was the Soviet reaction to the "Able Archer" exercises in the 1980's. Soviet intelligence had been operating under the assumption that American government was planning to go through with a nuclear strike on the USSR, with an internal codename of "RYAN". This believe originated from the top-down, rather than from the bottom-up, with both Breznev and Andropov (the head of intelligence and future leader of the Soviet Union) instructing their subordinates to look for signs of this plan and begin measures to prevent it coming to pass. The Able-Archer exercises came on the heels of the Pershing II deployment to Europe and the announcement of the SDI, and were interpreted through this lens as being a likely candidate for the anticipated nuclear strike. As a result, Soviet forces were played on high alert and made preparations for their own preemptive strike.

As to your question more directly, the The Sword and the Shield mentions the following conspiracies (from memory - they may mention more but I don't have the pages in front of me right now):

• AIDS being created by the American government. This conspiracy was entirely fabricated according to Mitrokhin.
• JFK assassination conspiracies. This was not invented by the KGB or people working on their behalf, but was promoted strongly by them. The book states that this includes the forging of letters purporting to be from Lee Harvey Oswald in an effort to implicate others in the assassination.
• Hoover as a homosexual. I do not recall the specifics of this one.
• Martin Luther King was the centerpiece of multiple conspiracies here. Initially, the KGB active measures regarding King attempted to promote the belief that he was working with the US government to tame the civil rights movement. This was because the KGB believed the MLK was potentially going to defuse racial tensions that they hoped would intensify to the point of open conflict. Later, after MLK died, the active measures shifted gear towards promoting him as a martyr killed by the US government. The motivation was the same in both cases, despite contradictory stories.

As to the question of "how" these were done, it varies. For the AIDS conspiracy, Jakob Segal was implicated in creating and promoting this. Segal was a Russian born East German professor of Biology. In the case of the JFK conspiracies, write Mark Lane is named as somebody that the KGB encouraged to publish information. In the case of MLK conspiracies, these were generally promoted in newsletters or publications that the KGB had some association with, directly or indirectly, and these were not necessarily American publications. Globally, there were a variety of outlets and organizations sympathetic to the USSR, or at least interested in uncovering the wrongdoings of the United States. The KGB was interested in funding and promoting these groups, even though the people and organizations involved were not always aware that they were on the KGB payroll, and the people espousing these theories were not necessarily speaking in bad faith just because they were being promoted by the KGB.

I'm hoping that there's somebody lurking with more detailed information here, because it's of interest to me as well!