What role did Napoleon the third play in Cinco de Mayo?

Answer 1

He was the mastermind behind the Second Franco-Mexican War (1862-1867):

As a consequence of the large indebtedness acquired after both the Mexican-American War (1846-1848) and the Reform Wars (1857-1861), the Mexican government faced such economic difficulties that president Benito Juarez suspended interest payments to foreign countries on July 17th, 1861. This angered France, Spain and England, who united their efforts to apply gunboat diplomacy on Mexico, and force it to continue with these payments. On December 8th of the same year, the Spanish fleet and troops from Spanish-controlled Cuba arrived at Mexico's main Gulf port, Veracruz.

Juarez negotiated the terms of such payments at the "La Soledad Preliminaries"; while Spain and England were satisfied and pulled their troops from the region, France wasn't interested in such debt: they were interested in conquering Mexico to prevent the rising of the United States as a world power by establishing a foothold in the Americas, and supporting the Confederate forces during the American Civil War (1861-1865). As a secondary objective, France was interested in exploiting the large silver, copper and iron mines located on northern Mexico.

On March 6th, 1862 5,000 French troops arrived to Veracruz, letting know friends and foes alike that France was in for the conquest of Mexico. The United States weren't able to protest until May 6th of 1862, but by that date, the legendary Battle of Puebla or Cinco de Mayo (May 5th, 1862) had already taken place.