This achievement is the third and final in a series of side missions.
There are a lot of spoilers contained within, because extensive RPGs like this have so many interlocking facets that it takes careful explanation to detail exactly where things can go wrong. Attempting to be unspecific in the pursuit of avoiding spoilers ultimately means you're going to stumble past the point of no return and never realize it.
The entire civil war questline will not suffer any foreseeable problems up to the campaign quest Alduin's Bane, where you must get an Elder Scroll and read it at a time wound at the Throat of the World. There is a quest called Season Unending that takes place concurrently with the next story quest, The Fallen, which for reasons unbeknownst to me makes War Hero missable if completed but does not affect the entire rest of the civil war questline.
Therefore, the appearance of the quest Alduin's Bane means that you need to direct your full attention to the civil war and complete the War Hero achievement before doing any more campaign missions. Hero of Skyrim can be done at any time because the civil war will pick up right where it left off after you complete the campaign.
This achievement technically begins all the way at the beginning of the game. Right after the dragon attacks Helgen and interrupts your execution, you can either follow the Imperial soldier Hadvar or the Stormcloak Ralof. This choice does not permanently slant your decision from the get-go, but the NPC you choose does give you a quest marker to go join their side of the civil war. You can still join the other side, and it seems the only difference of your choice of NPC at this point is that Hadvar becomes essential and frequently joins you for civil war missions if you follow him out of Helgen and later choose to be Imperial.
You still have to actually pick a side despite what you chose above. To make it easier for you, both sides are exactly equal in terms of quest rewards and their effect on the game world. To join the Imperial Army, head to Castle Dour in Solitude and speak to General Tullius, who will direct you to Legate Rikke. To join the Stormcloaks, go to the Palace of Kings in Windhelm and speak to Ulfric Stormcloak, who will send you to Galmar Stone-Fist. Each will have an initiation task for you to complete, both ending in an oath of loyalty to the faction and the Taking Sides achievement.
The only significant change in the game world after choosing a side is that the other side considers you an enemy. Approaching camps or forts owned by the side you did not choose will cause soldiers to yell at you and warn you to leave, and will attack if you do not.
Regardless of the side chosen, you will have to travel to Korvanjund to obtain a relic called The Jagged Crown, then deliver a message to Whiterun shortly before being tasked with repelling an attack on the city from the opposite faction.
At this point, both sides will task you with taking a series of forts. I believe it is five total forts that must be taken with the third one being Fort Greenwall for the Imperial Army or Fort Sungard for the Stormcloaks. There is absolutely nothing special about these particular forts as compared to the rest of them you're required to sack, except that you receive the War Hero achievement for completing the quest.
At the end of the civil war questline, you will finally take the base city for the opposing faction. You will attack Windhelm if you are Imperial and Solitude if you are Stormcloak.
There will be infinitely respawning enemy soldiers as well as tons of wooden barricades littered about to stall your progress. The barricades can be destroyed if you're not feeling like running the rat maze the enemies have set up for you. Your goal is the keep for the opposing commander, being Castle Dour in Solitude or Palace of the Kings for Windhelm.
A short cutscene will play as the commander of the faction you chose attempts to talk down the commander of the faction you didn't. It never succeeds and you are put into combat with the opposing faction's lieutenant and commander. The lieutenant must die (Galmar Stone-Fist if you're Imperial, Legate Rikke if you're Stormcloak) and you have to beat on the commander until they fall to the ground (Ulfric Stormcloak if Imperial, General Tullius if Stormcloak.)
At this point you are given the option to perform the death blow on the enemy commander regardless of which side you choose, and if you choose not to do it yourself then the opposing commander does it for you. The civil war is now over and and you receive the Hero of Skyrim achievement at this point.