How did Charles de Gaulle become the leading figure of French liberation movement?

TL; DR :
De Gaulle was one of many people who favored continuing the armed resistance against the Nazis. Being less known and less important served him well, as he could flee to Britain without being accused of deserting his country or reneging on his responsibilities. In the same time, once in London, he was protected from the Vichy government, which moved swiftly to neutralize all possible opponents in the French mainland and the colonies (by imprisoning them or engaging/coercing them into collaboration).

June 16, 1940
This is the last day of Paul Raynaud's government. As it is clear that the party favoring the armstice with Germans is about to come to power, general Edward Spears, the trusted representative of Churchill to the French government, is looking for an influential French politician who could come to London and from the French government in exile, following the example of the Czech and Polish government at the time. Spears notably approaches Raynaud (the outgoing prime minister) and Georges Mandel (the minister of the interior) - both refuse. Next day Spears leaves for London with de Gaulle, who however at that point is not seen as such a figure.

Mandel
Mandel was apparently the Churchill's preferred choice, as he was a consistent and vocal opponent of Nazism since 1933 (while Raynaud was blamed by Churchill for defeatist attitude and sadden swings of mood). A few days later the British ambassador again offered to Mandel to flee to London on a British destroyer. Mandel refuses, saying that such a move would be seen as a defection, and that he intends to continue fighting from the French colonies. He will then travel to Africa on board of Massilia, be arrested there, imprisoned, handed to Germans and eventually assassinated.

Massilia
27 members of the parlament, including Georges Mandel, Eduard Daladier, Jean Zay, and Pierre Mendès, tried to reach the French colonies in Africa, in order to organize the resistance from there. Because of a chain of unfavorable events and interference from Pétain government, the boat sails only on June 21, i.e., on the eve of signing the armstice. They arrive to Casablanca a few days later, at which point the local French authorities are inclined to side with Pétain government: most of the passangers are arrested and returned to the mainland.

The process of Riom
The high-profile figures from the Raynaud government and the army, including Raynaud himself, Mandel and Daladier, are judged for plunging the country into the war.

De Gaulle
De Gaulle joined the government on June 5, less than two weeks before its fall. Although officially an undersecretary for war, he was de facto the war minister, since the post was vacant (more precisely, Reynaud acted as his own war and foreign ministers). In practice de Gaulle was spending most of his time shuttling between France and London, acting as atrusted representative of Raynaud, i.e., the French equivalent of general Spears. Thus, de Gaulle was highly placed, while lacking potentially harmful political luggage - the fact that was fully understood by those who promoted him. As Mandel said in his pep-talk to de Gaulle² (my translation) :

"In any case, we are just at the beginning of the world war. You have big things to accomplish, general! But with an advantage of being, among us all, an untarnished man. Think only about things that need to be done for France and keep in mind that, when necessary, your current role could only ease things."

De Gaulle left France on the first day of Pétain government in the Spears' airplane, with a significant sum of money given to him by Raynaud, and a limited assignment to make a counter-announcement after Pétain declares that his government seeks the armstice. As de Gaulle himself remarked¹:

"That's good. They don't want me! I get out of here to London."

By "they" he could have meant both Pétain, for not offering de Gaulle a place in the government, and Raynaud and Mandel, who were removing him away from where the main action was supposed to take place.

To summarize:

• De Gaulle was well placed and with untarnished reputation
• There was no shortage of brave and capable men who were willing to continue the war, but they were successfully tracked by the Pétain government
• de Gaulle's low profile worked in his favor: his reputation was untarnished and he didn't have political responsibilities that would prevent him from leaving France.

Footnotes

1. "De toute façon, nous ne sommes qu'au début de la guerre mondiale. Vous aurez de grands devoirs à remplir, Général! Mais avec l'avantage d'être, au milieu de nous tous, un homme intact. Ne pensez qu'à ce qui doit être fait pour la France et songez que, le cas échéant, votre fonction actuelle pourra vous faciliter les choses."
2. "C'est bon, ils ne veulent pas de moi ! Dans ces conditions, je fous le camp à Londres"

A large part of the issue was that de Gaulle (born 1890) was one of the few French leaders who was the optimal age (around 50) for generalship and national leadership. Foreign contemporaries in this group include America's Dwight Eisenhower (1890), Britain's John Verreker, Lord Gort (1886), and Germany's Erwin Rommel (1890). Adolf Hitler himself was born a year earlier, in 1889.

On the other hand, the French leaders were a geriatric bunch. Politicians included Leon Blum (1872), Paul Reynaud (1878), Pierre Laval (1883), and Eduoard Daladier (1884). Among the French military, names and birth years included Jean Darlan (1881), Henri Giraud 1879), Maurice Gamelin (1872), Maxime Weygand (1867), and the "grand old man," Phillippe Petain (1856). Only Jean Lattre de Tassigny (1889) was around de Gaulle's age. Marc Bloch (1886), who also deserves mention as a historian and resistance fighter, was not as strategically placed as de Gaulle.

De Gaulle had a head start on his age group, because their "places" were occupied by the other men above. And when the older men left the scene, they created a vacuum for de Gaulle to fill. Neither individually nor collectively were the older men capable of leading the "liberation."

Source: Wikipedia, various bios.

I don't know if that counts as a full answer, but it might be worth more than a comment.

De Gaulle was one of the French proponents of armored warfare. France lost partially due to its insistence on diluting tanks amongst infantry units, which De Gaulle had argued against. On the other hand Germany won precisely by following the type of warfare that had been theorized by De Gaulle among others (Liddell Hart, Tukhachevsky and Guderian...). Being on the side of those whose new theories had been proven successful beyond all expectations can't have hurt, especially when compared to more senior officers whose outdated thinking led to such a calamitous defeat.

Speaking of Giraud:

When World War II began, Giraud was a member of the Superior War Council, and disagreed with Charles de Gaulle about the tactics of using armoured troops.

And furthermore, Giraud was a POW until his escape in 1942, so he couldn't have picked early on.

Also, there was a dramatic element to De Gaulle's speech about continuing the war on June 18 and I guess that counted for a lot to make him the symbol of the continued war, after the speech.

De Gaulle announced his intention to broadcast again the following evening. He was furious to discover that his historic broadcast had not been recorded, as BBC engineers with limited equipment had failed to recognise the importance of the speaker or of his speech. On the 22nd de Gaulle broadcast again, and repeated his message in a speech that was heard much more widely. This time it was recorded. De Gaulle was recognised by Churchill as "the leader of all Free Frenchmen, wherever they may be" and made many more broadcasts to France.

As far as 1961 goes, I am surprised that is being considered. De Gaulle was brought back to "fix" Algeria and his "fix" consisted of hacking the Gordian Knot and declaring his intention to leave Algeria. No surprise the military elements who wanted to continue the war weren't impressed.

Why did Churchill choose de Gaulle to lead the French resistance? (and was it really up to Churchill to decide?)

Churchill could decide among what he had available. De Gaulle was in the UK, had the will to continue the war, had shown that he had initiative. He had got some fame during the war.

Here his relatively lower rank could even be an advantage; a more senior officer or politician would have been tainted by his role in the failure of France in 1940. De Gaulle was low enough that it could be presented as the military leader that could have saved France if he had been given the opportunity to.

Of course maybe you could point to "but Mr. X had all of this, why did Churchill chose De Gaulle?" Well, there were not so many of the candidates, and Churchill had to select one. In fact, for some time it was not clear that De Gaulle was the only candidate (see Darlan, for example).

In any case, and after propping up De Gaulle's figure through the BBC for a long time, changing the "Free France" leader could easily backfire. It could make him look like a puppet, and that would reduce its usefulness.

The relationship with Petain is a red herring. He had had a relationship with Petain while Petain was the hero of Verdun. After Petain became head of Vichy France, De Gaulle was opposed to him. Why would Petain's actions as head of Vichy France stain De Gaulle's reputation^*1?

Also Moreover, the allies continued treating Vichy government as the official one for several years after. So they were playing with several actors to see which one was more convenient to them. So what? That is politics 101. De Gaulle had to put up with that, which shows who was calling the shots.

Why did the French in France in June 1940 - June 1941 rally behind de Gaulle ?

Did they? Some did collaborate with the Germans. Many fought for Vichy France. Others fought for the communists. Most tried to survive^*2.

But in 1945 a "France united under De Gaulle's leadership" was a nice tale that was very convenient for forgiving those who did not fight (Vichy), for forgetting about those who did fight but did not win (Communists) and for De Gaulle.

Why did they continue to support him afterwards, despite his alliance with the communists, the recent armstice supporters?

For many people it was either to support him, the communists or the Germans. And he was propped up by the UK and USA armies.

Also De Gaulle had to make do with what he had available. De Gaulle did not "give the Communists the leading role in the Resistence"; after Germany invaded the Soviet Union, the communists became the leading force behind the Resistence (the first German officer killed by the resistence was killed by the communists, more than a year after France had been defeated). That title was not something "given", it was something earned by their activities and risk.

And in any case, you seem to overestimate the importance of the Resistance. It did help, but it was from being a decisive factor. But again, it was a nice story to justify the tale that France did not "actually" surrender.

Why did French military eventually submitted itself to de Gaulle?

The combined power of the UK, the USA and the Soviet Union seems like a convincing argument to me. And probably some of them were happy to submit to a French leader (De Gaulle, or perhaps any French leader) who got them rid of the Germans.

^*1 Also you should differentiate about Petain's reputation during and after the war. After the war it was easy to say "fight to the last man" and repudiate any appeas*m*nt/collaboration towards Germany, during the war "trying to preserve something of France after its crushing defeat" was probably a more popular option.

^*2 Again here you seem to have a somewhat romantic, heroic vision in the past. I will take some offense on your I don't like its hidden premise that the remaining 30 million French were somehow all cowards/collaborators comment.

It is sooo easy to see a patriotic film and say "freedom or death". But guess what? Most people want to survive (themselves and their families). In additon to that, the French of 1940 had seen how their army, considered one of the most powerfull in the world, had been crushed in two months. It is not an scenario that entices untrained, unarmed civilians to start a rebellion, isn't it? If you feel that they are "cowards" for that, maybe you are very brave. Or maybe you do not know what it means under the occupation of a foreign army.