I understand that Riften was wrecked before the events of Skyrim and it took 5 years to rebuild the City. How instrumental was the Black-Briar if they can even claim any legitimate credit to rebuilding of Riften?

Riften was burned down in 4E 129, and was rebuilt to roughly where it is today by 134. According to Hemming Black-Briar, the family's meadery was built almost fifty years prior to the game's start in 4E 201. Meaning that the family didn't actually have a major presence in the city until some time close to 4E 150. That means that while they probably boosted the city's economy in a major way, the Black-Briars probably had little to no role in the actual rebuilding process.

So they probably got into power through underhand means and got rich together with the rebuilt city.

Or they built it legit then got corrupt later.