The third republic was created after the fall of Napoleon 3, in 1871. At the time, only 1/3 of the MPs were republicans, the other monarchists, and a third restauration was possible.... and failed because the monarchists couldn't decide who should be king (it is a simplification). Mac Mahon became president, for a seven year mandate, because the monarchists thought it would be enough for the legitimist pretendant to die (he was old and childless), thus uniting the monarchists under one banner.... it didn't happened. The republic became strong.... the president became an elected symbol of the country, and the prime minister, relying on a majority in the parliament, became the "strong man". The 3rd republic was a parliamentary regime.

During the thirties, the right wing became less interested in a monarchic restauration, and more in favour of a presidential regime. Just before WW2, the PSF (nationalistic right, catholic) was rising quickly: their alliance with the radicals (center, republicans) became stronger... but WW2 and the support (ambiguous: De La Roque professed a loyalty to Petain, but entered the resistance) offered to Petain by some members of the PSF had a huge impact on the influence of their ideas (among them, the presidential regime). At the end of WW2, the new constitution created a regime (4th republic) similar to the one in existence before war (3rd republic).

The Gaullism was the ideological successor of the PSF. De Gaulle had quite an aura in France. Although the social and economic situation of France during the decade after the war was good, the Algerian war and general unrest in the colonies, along with a certain unstability, had huge political effects. In 1958, several high ranked officers, fearing that France would abandon Algeria as it had Indochina, launched a coup. From Algeria, they invaded Corsica, asked for De Gaulle to be named Prime minister and be given extended powers to deal with the Algerian situation, and prepared an invasion of metropolitan France. De Gaulle was named as Prime minister by President Coty, and given these extended powers. He used them to draft a new constitution, creating a presidential regime (although the PSF did not liked De Gaulle actions during the war, he shared a lot of their ideals), and this new constitution obtained 80% of Yes in the following referendum (De Gaulle popularity, and the 4th republic political instability, helped). Thus was born the 5th republic, a quasi-presidential regime (it is, most of the time, except in the rare circumstances where the majority of the parliament is not from the same party as the president, which may not happen anymore since the mandate of the president is now only 5-year long, as is the one of the MPs, and the elections take places only a month far one from another). By the way, 4 years later De Gaulle did negociate the Algerian independance, the dreaded thing that drove Salan and a few other officer to put him in the presidential seat...

TLDR; De Gaulle did indeed change the roles of the president and of the prime minister, following a kind of military coup. Then did what the military thought he wouldn't.