YEAH! This is my type of thread!
So, I'll assume that I am leading Skyrim following a successful revolution.
I would happily talk with the Empire, would even rejoin it on the term that Skyrim's Internal Politics are not interfered with by the Empire, and that Skyrim was placed in a position where, when the Empire does collapse, the aftershocks do not reach Skyrim. This could be done by demanding a large sum of gold from the Empire to rebuild and improve Skyrim's infrastructure. In return, Skyrim would pay tax to the Empire, imperial Troops would be permitted Garrisons withing Skyrim, and Skyrim's forces would come to the aid of the Empire if needed. Abusing the right to independent politics, I'd immediately begin talks with Hammerfell to find out where Skyrim's strongest allies are, and, if possible, create closer ties with Hammerfell.
In this scenario, Skyrim is part of the Empire, yet independent and strong. With ties to Hammerfell, we would be able to keep the Empire alive, but pressure it to move further away from the Aldmeri Dominion, and perhaps end in a scenario where it is run by one of its own client kingdoms!
If that plan failed (which given its idealism, it very well could), I'd move straight for ties with Hammerfell, while either appealing to the North Cities of Cyrodiil to join Skyrim, or directly annex them (after a while, that is. let the Empire die a bit first). in the case of bruma, I'd incite a Revolution amungst the Local Nords, if it hasn't already erupted, in the hopes of bringing it into Skyrim (as it is a key defensive center).
In the absolute worst case scenario and Hammerfell decides to leave me hanging, I'd focus on taking these northern cities for myself, and prepare for all hell to break loose. In either scenario, I'd use the additional income from these cities to rebuild and improve nordic infrastructure.
At home, the situation would remain the same either way, rebuilding. In particular, I'd utilise Imperial POWs and captured Thalmor Justiciars as a sort of slave labour force (In a rather loose use of the term), using them as cheap labour. In the case of Imperials, I'd have them build settlements which, when completed, they will be permitted to settle as Nordic citizens (ultimately adding to Skyrims agricultural output). In the case of the Thalmor, well, I'd be less forgiving. Think Cidhna mine filled with Justiciars.
In attempts to rebuild the army, I have an innovative approach. Training will consist of basic recruit training, followed by sparring with fellow recruits and eventually more experienced warriors. FInally, recruits will be put to the task of clearing bandit camps (under more experienced leadership of course) in an attempt to A. Give them a better feel for what it feels like to face an opponent in a life and death situation, and B. To help clear trade routes.
Markarth is a different issue. I would weigh up costs, and determine if it is worth wagin war against the Forsworn or not. if so, then i'd do as follows to try and curb their power:

The Ridge between the Reach and Whiterun is crawling with Forsworn camps, and most notably, a number of redoubts. Controlling these redoubts provides greater control over the reach, and makes it virtually impossible for the Forsworn to conduct a conventional war. From there, I'd attack and occupy as many redoubts in the Reach as I can, notably Dead Crone Rock, Deepwood Redoubt, and Lost Valley Redoubt. The Karthspire would be taken, and granted to the Blades in return for them agreeing not to harm Skyrim or Hammerfell (but let their fury loose upon the Thalmor).
Of course, this military campaign must work with a civil campaign in Markarth itself. Greater welfare would be grated to the inhabitants of the Warrens, and granting Reachmen land in return for service may also be beneficial (although granting them land may backfire, causing them to revolt against Skyrim again).