For starters, the POD would have to be after 1958, since in the IVth Republic the president was a figurehead with a short shelf life, because of chronic governmental instability.
One possibility, for what it's worth, would be the POD I once called "A Hundred Magic Bullets". On August 22, 1962, a cell from the terrorist organization OAS (created by radical Pieds-Noirs to fight against Algerian independence) ambushed de Gaulle's motorcade and opened fire on his car with automatic weapons. In OTL, although some 150 rounds were fired, none of them hit and de Gaulle escaped unharmed. So, have one bullet find its way to its intended target, and France finds itself without a president.
In such circumstances, the chairman of the Senate would have become acting president pending elections (in 1962, that man would have been
Gaston Monnerville, incidentally a colored man from Guyana). One could imagine that the chaos caused by de Gaulle's assassination, coming on top of the trauma of defeat in Algeria and the messy repatriation of over a million European settlers, might somehow lead to elections being delayed for three years. At that point, Mitterrand, presenting himself as the natural alternative to the political status quo, could get a majority of the vote (in OTL he got 44% in the run-off against de Gaulle, which was considered a honorable score given the latter's popularity).
As to how and why he'd stay in power such a long time, I don't really have an idea. There are no formal term limits in the Vth Republic, but the voters would probably tire of him after his second term.