Charles de Gaulle openly detested the new world order of the Cold War. He disliked superpowers and did not appreciate American hegemony over Western Europe. His primary foreign policy aim, then, was to make France an independent counterbalancing force—making Western Europe relevant again as an independent entity by becoming a leader in European politics. de Gaulle believed that while France had declined, its day had not yet passed, and it was destined to rise again. His foreign policy would thus encapsulate some of the major tenets of what is known as Gaullism.

de Gaulle clashed with the Americans primarily because he perceived them as the more dominant force of the bipolar world. Many of his policies have been interpreted as anti-American, but other actions—such as his support for the Americans during the Cuban Missile Crisis—reveal that he was just as much anti-Soviet. France's antagonistic relationship towards the Americans was a direct consequence of de Gaulle's desire to enhance France's independent stature in a world dominated by superpower politics.

With specific respect to NATO/OTAN: NATO was created in 1949 as a defensive alliance among Western states to oppose the Soviet Union and the countries east of the Iron Curtain. When de Gaulle inherited the NATO relationship during the fifth republic, he argued that France deserved an equal place at the NATO table with the U.S. in policymaking. More specifically, he didn't like the idea of the U.S. holding unilateral power to authorize nuclear force. When de Gaulle proposed that France and Britain should have an equal say as the U.S., he was rebuffed by President Eisenhower. This was in 1959, but it put the writing on the wall for France's 1966 departure. Immediately following Eisenhower's rejection, de Gaulle removed the strategic French Mediterranean fleet from NATO. In 1963, he signaled his intentions even more clearly by saying that military cooperation had had its day. And in 1966, France fully withdrew.

de Gaulle also made it a priority to create a nuclear weapons stockpile that could independently function as a deterrent, and so France was able to reject the U.S. nuclear umbrella in favor of establishing its own program. In 13 February 1960, France tested its first nuclear weapon in the Sahara Desert, becoming the fourth country in the world to do so. (And France today has the world’s third-largest stockpile).

Even though France was nominally protected under the U.S. nuclear umbrella, de Gaulle wanted to develop his own deterrent anyway, as doing so would guarantee French sovereignty, allow France to assert its own interests, and ensure the retention of its permanent seat on the UN Security Council.

Source: Conklin et al., France and its Empire Since 1870

It stems from a basic difference in views about France and France's role in the world... and in no small part is due to the dislike between Charles de Gaulle and the US and British leaders.

To the US, France was a leading partner in Europe, like West Germany, Britain, and Italy. But they were a junior partner. The US was the Super Power and should be given appropriate deference.

De Gaulle disagreed. He thought France was and should be a great power in its own right. He could read, so he could see that the US had more people and a larger economy. But he didn't think France needed to relegate itself to a lesser tier. In his mind France had an independent seat on the UN Security Council, it had its own nuclear deterrent. And he thought France could have its own foreign and military policy, thank you very much. He didn't leave NATO as a parter, he just left it as an active one. If the Soviets attacked, France would be there. But until then there was no need for a permanent American presence.

And this was part of a long term plan for de Gaulle. With just West Germany to deal with, the Common Market could be dominated by France. And France was going through boom times economically. The long "vision" was a stable Western Europe (the EEC 6: France, W. Germany, Italy, Belgium, Netherlands, Luxembourg) that would be dominated politically and economically by France. The EEC was going to be France's Commonwealth. And while de Gaulle needed a friendly Britain and US. He didn't think he needed them on the continent. And withdrawing from the NATO command structure was seen as a way to hopefully distance the EEC and France from the Atlantic powers.

Don Cook covers it in Charles De Gaulle Biography

It should be noted that France did not leave NATO, but only withdrew from the integrated command structure.