The Cold War started soon after World War II. It started as the Soviet Union and the Western Allies (particularly the United States) started to compete for domination of the post-war world.
As WWII was ending, the Allies split Europe between themselves, with the Soviets gaining control of Eastern Europe. They also divided Germany. But the two sides did not trust one another. This was because their political/economic systems were completely opposed to each other with each side thinking the other's system was evil. The two sides quickly became suspicious of each other. Each side thought the other was pushing for world domination. These tensions caused the Cold War to begin.
It is impossible to know WHEN the Cold War truly began since its origins are highly disputed between. Most regard the 1942-1945 period as the beginning of the Cold War. I agree because the Soviets became highly sour when the US and other allies did not come to the Russian aid as Hitler was dessimating Soviet territory.
Others take the Potsdam Conference as the beginning of the war since the US began 'tryinng' to dictate Soviet actions towards the treatment of their satellite states.