In 1862, Mexico was in financial crisis, and as an emergency measure the government temporarily suspended payments of national debts to its European creditors. Three European governments decided not to stand for this, and troops from Great Britain, Spain and France were landed at Veracruz to coerce payments. The British and Spanish managed to secure a few concessions, but it soon became obvious to them that Napoleon III, Emperor of France, had much larger designs in mind and was merely using the debt crisis as an excuse for further colonial expansion. The British and Spanish wanted no part of this and pulled out their troops, but the French brought more and more, beginning the French Intervention in earnest.
Napoleon III was gambling that while the United States was busy in a civil war, he had several years to act freely without American interference. He was correct - the best Lincoln could do until 1865 was send scathing letters and make threatening declarations. However, as the Confederacy crumbled and Federal troops became available, the US Army began collecting forces in Texas. By late 1865, this force numbered around 80,000, General Sheridan was in command of it, and the rhetoric from President Johnson and many Congressmen had grown quite belligerent. The US wanted the French out of Mexico, and was ready to invade to force the issue. This campaign assumes the invasion becomes necessary.

The Republicans

The French intervention drove the democratically elected government of Mexico out of the capitol, and it spent years moving between hiding spots in the vast Mexican countryside. Benito Juaréz remained president throughout this period, and remained a primary linchpin holding the Republican cause together. He came very close to capture in 1865, shortly after the surrender of Lee's army at Appomattox, while the rump Republican government had its back against the Rio Grande and the French army hunted it.

The French Army

The French army in Mexico consisted at first of colonial troops and light artillery, but over time grew into a colorful array of light and line infantry, light and heavy cavalry, zouaves, and colonial battalions from Africa. The French Foreign Legion was present throughout the era, and experienced it's defining moment at the Battle of Camerone in 1863, when a small detachment died almost to a man fighting off hundreds of Mexican Republican Army troops.

The Emperor of Mexico

In 1864, a cabal of Mexican aristocrats "invited" Maximilian, the younger son of the House of Habsburg, to ascend the throne of Mexico and rule as Emperor. Maximilian was educated, intelligent, optimistic, and embodied a peculiarly 19th century mix of old world traditions and liberal ideals. He tried in earnest to form a loyal and stable government to rule his adopted country, but couldn't get much popular support, and had to lean heavily on French military support. With Napoleon's help and encouragement, Maximilian formed a loyal Imperial army, but to fill the ranks he had to recruit heavily from abroad. His army included whole units of Austrian and Belgian adventurers, and a few ex-Confederates starting in 1865. With greater resources or the right enticements, there could have been even more foreigners among the Imperial forces.

Confederates

Thousands of Confederates fled to Mexico as the Confederate cause crumbled, and Confederate general Jo Shelby famously brought his entire brigade of cavalry to serve the Emperor. Maximilian decided not to bolster his army with thousands of exiled American combat veterans, but might have changed his mind in the face of a US invasion.

The US Army

The US Army had just won the Civil War, and many period observers thought its artillery arm to be the best in the world. Years of hard combat had given it a large cadre of experienced officers, a firm grasp of operational logistics, and a hard core of veterans. In 1865, whole corps were sent to Texas to organize for invasion in case war with France became necessary. This army was a mix of veterans and new recruits, including many thousands of recently freed slaves. It was commanded by Philip Sheridan, one of the more determined and effective US generals of the Civil War, and a man with a firm grasp of logistics and grand strategy.

The Country

Mexico is a huge country, most of it in the tropics and sparsely populated, and the French found it very difficult to operate across its huge desert wastelands, numerous rugged mountain ranges, and vast tropical jungles. Even a brief study of the Mexican American War of 1845-48 shows that an invading US Army would have as well. Both the Imperial and Republican camps of Mexico used guerilla tactics, which would have complicated formal military operations immensely. The tropical lowlands were notoriously unhealthy for Europeans and Americans, wiping out entire armies stationed too long in the breeding grounds for yellow fever and malaria. France depended on supply by sea and a US army in Mexico would have too, necessitating strong garrisons in tropical lowland ports.

The Navies

The US Navy was the most numerous in the world in 1865, but vied with the navy of France for the rank of second most powerful (after the Royal Navy of the United Kingdom). Both contained a wild mix of hull designs from traditional wooden ships to newfangled ironclads, equipped with many different types of artillery and steam propulsion. Most US Navy units were small and meant for coastal operations, while the French navy was mostly a blue water force meant for operations supporting a global empire. The contest between these two mismatched forces over the Mexican Gulf coastline would have been difficult, dramatic, and full of oddities.