This is vastly more complicated than I'm making it out to be:

I'll start off by mentioning the fact that the Belgians didn't really prepare Congo for independence very well: the number of college graduates in the newly independent country were in the teens. This was a recipe for disaster, particularly in a sprawling country with vast natural resources and hundreds of ethnic groups and languages, who had no concept of the idea of a "Congolese". The country was completely undeveloped.

So, in this context, a guy named Patrice Lumumba was the first prime minister. Within a few months of independence, various secessionist movements predictably begin. The USA and the Soviet Union supported different factions: as mentioned, the riches of the Congo made it a battleground in the Cold War. Lumumba decided to solicit help from the Soviet Union. This led to a coup by a guy who would eventually become known as Mobutu Sese Seko. He expelled the Soviet advisors and executed Lumumba.

Mobutu would rule until 1997, proving that no matter how bad things are, you can somehow make it worse. His anti-Communism guaranteed support from the West and the IMF, and later China, throughout the Cold War, given the Soviet learning orientation of most of his decolonized neighbours during the time. The one thing that can be said about him is that he basically kept the Congo somehow together, a very tall order. Mobutu, however, was infamous for his increasing outlandishness during his rule-during the early 70s, he began patterning a personality cult based off of those of Mao and Kim Il Sung. For a time, nobody was even allowed to mention anybody but him on the news on state TV, and the evening news started with an image of him descending from the clouds. (Funny quote from Mobutu's visit to Beijing when relations improved. Mao-"I spent a lot of money and arms trying to overthrow you." Mobutu-"Well, you backed the wrong man.")

Apart from widespread human rights violations, Mobutu was grossly corrupt. Flying Concordes to Paris for shopping trips with funds from the state treasury level corrupt-Mobutu's decadence went beyond parody levels. (Check out the wedding of his daughter, Yaki. Or the fact that Mobutu married twins.) If you were a tax payer in the United States during the 60s, 70s and 80s, chances are your tax dollars went to fund some of his extracurricular activities. All of our Presidents during that time-Kennedy, Johnson, Nixon, Ford, Carter, Reagan, Bush Senior-came to the same conclusion, that the risks of overthrowing Mobutu were not worth the payoff. Mobutu also created a system where every government official depended on corruption to earn money. The bureaucrats pretended to work, and the state pretended to pay them, basically.

By the 90s, the state was a complete economic basketcase, in part due to thie. The end of the Cold War meant that the USA began to distance itself, under Clinton, from Mobutu-he would be, along with Suharto, the last of the old style major Cold War dictators to go. Due to the general unrest in the country caused by the economic problems, Mobutu agreed to sham elections around 1994. However, this would be in vain due to what was going on in the east of the country. In 1996, Mobutu tried to expel the Tutsis-the target of the Rwandan genocide, and Paul Kagame's ethnic group-from the country. He was also sheltering many of the Hutu soldiers who had committed crimes against humanity during the Rwandan Genocide. Tensions blew over, and Rwanda and Uganda supported Kabila, who was the head of the rebels. In spite of the seemingly pathetic size differential, the Congolese Army, unpaid and unmotivated, completely collapsed. Mobutu was by this point ailing, physically and mentally. By mid-1997, they had nearly taken over the country, and Mobutu fled into exile, where he discovered that even he couldn't flee or bribe prostate cancer.

The white mercenaries were there because of the secessionist movements. Che was in Africa, but was very unimpressed with the Africans.