As soon as you win 3 championships in a period of roughly 5 years, they're crowning your team as having a dynasty. There's no set number of wins, over any specific period of time, it's just that you're winning the majority of them in a brief period.
The real, great dynasty title belongs to John Wooden's UCLA basketball teams, who won everything in sight for a number of years, and the New York Yankees, that dominated baseball for decades. The rest are peanuts by comparison, but still not as large in significance.
But, since there are no specific requirements, at this time, the Houston Astros baseball team has been a dynasty within their own division in baseball, because they finish at the top of it so consistently.
In the NFL, the one true dynasty was the Packers of the 60s. The rest are fine within their own right, but the Packers feat sits up there with the UCLA and Yankee feats.