What we commonly refer to as the "Allies" in World War 2 (mainly Britain, China, the United States, and the Soviet Union) in fact called themselves the "United Nations" during the war. One of the goals of the United Nations--and here we can see Franklin Roosevelt taking the lead in order to make up for what he viewed as the great "mistake" of the WWI peace agreement, the American refusal to join the League of Nations-was to maintain the alliance into the postwar era in order to bring a final stability to international affairs. As you can imagine, this made for a strange alliance at the time--the bastion of capitalism was allied with the Communist-led Soviet Union. A massive propaganda campaign, assisted by Hollywood, nevertheless tried to imprint in the minds of Americans that the UN was an absolute necessity not only in war but in peace (e.g., Mission to Moscow and other films referring to Stalin as "Uncle Joe"). It may also seem strange that China was included as one of the big-four within the UN, but it was already a massive nation and incredibly important internationally, though not to the extent that it is today.

France was a defeated power to be sure, but it's international importance, as well as its importance in maintaining peace on the European continent, found it a spot on the council (luckily Stalin was more concerned with preventing the Germans from invading Russia a third time than he was in preventing American allies from stacking the Security Council).

As /u/EllesarisEllendil explained, it was really that simple--the United Nations, in their victory, chose themselves to be the permanent members of the UN Security Council after the war (with four seats given to nations on rotation). The "Big Three" at the time were Britain, and United States, and the Soviet Union, and so you can see the negotiations for the UN at Dumbarton Oaks in 1944 (China was present as well), though negotiations over exactly what the UN would look like in the postwar era continued through the more-famous Yalta and Potsdam Conferences which only included the Big Three (at Yalta, Roosevelt conceded a "Soviet sphere of influence" in Central and Eastern Europe in exchange for Stalin's assurances that the Soviet Union would remain an American ally even after the defeat of Germany). He does simplify the situation with China, however. Even as the Second World War ended, China was embroiled in a civil war between the nationalists (led by American ally Chiang Kaishek) and the Communists (led by the infamous Mao Zedong). When China won the war in 1949, it is true that the Second Red Scare (McCarthyism) was in bloom in the United States, and that this played a role in the UN giving China's seat to Taiwan, the island nation (China has never considered it independent) to which Kaishek fled after his defeat on the mainland. An interesting side note--the USSR protested this move by abstaining from Security Council votes, and it is the only reason that the UN was able to authorize action in Korea in 1950. Anyway, the Red Scare was long over by the time the United States agreed that People's Republic of China should hold the seat on the Security Council. This was no easy decision; after all, the United States still held a military alliance with Taiwan. But Nixon rapidly improved relations with both the USSR the China after assuming power in 1969, and this was just one more step in the Chinese-American rapprochement that helped to define the period of "Detente" with the Communist powers (yes, the word "Detente" typically refers to Soviet-American relations).

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A great monograph analyzing the goals of the United Nations during the war, its tumultuous existence as war wound down, and the film campaign that the United States waged in an attempt to convince Americans and the Soviets that the alliance was natural and could last is One World, Big Screen: Hollywood, the Allies, and World War II (Chapel Hill: University of North Carolina Press, 2012) by Todd Bennett.

It was decided by the US, USSR, Britain and China at Dumbarton Oaks near Washington in 1944 (http://www.ibiblio.org/pha/policy/1944/441007a.html), and incorporated in the 1945 UN Charter. The US apparently proposed at Yalta to add Brazil, but the British and Soviets opposed.

It's basically a reworking of the inter-war League of Nations Council, originally to be Britain, France, Italy, Japan and the US - though the last never became a member - with four non-permanent members (raised to six in 1944 and ten since).