What goes on in Paris for the Tour de France final?

People line the streets from the entry point into the city all the way up to the finish line on the Champs-Elysée.

It will be so crowded on the Champs that there's no point in even going unless you know someone with an upstairs apartment or you want to get there at dawn and wait all day. If you want to watch, it's better to find another point on the route in from the edge of the city.

Street traffic of course will be a mess because of road closings on the race route. Expect delays on Line 1 of the metro, and the stations along the Champs-Elysée will be completely closed.

EDIT: The route doesn't come close to Montmartre this year. They will enter in the southeast from Porte de Charenton or Porte de Bercy, follow the north bank of the river for a bit, cross the river, follow the south bank of the river for long while, cross the river again to Place de la Concorde and the Champs-Elysée.

Personally, I would find a place in one of the parks along this route. There's a pedestrian walkway on the south edge of the Parc de Bercy that might provide some high ground, or the steps on the north side of the François Mitterand national library. A little farther downroute, there's the large Place Valhubert on the north edge of the Jardin des Plantes, or the open air sculpture museum along the Quai Saint Bernard.