Benefitted enormously from Mercedes engine and rear-end, yet made most of it.
Interesting to remember that their first technical partnership was actually with McLaren in 2009. Basically this is what laid the foundations for their rise. Not only got they Mercedes engines - first time Merc' started to supply a customer team in F1! But they also bought the whole transmission. I think Force India may have been the first one, who went that route? But they set off in this way and soon it became standard for customer teams to purchase the whole drivetrain.
Beginning of 2009 was like the old MF1/Spyker/FI, who was at the back. But their in-season development was one of the best ever seen, up there with McLaren that year. Interestingly, 2009 Belgian GP has remained their best ever chance to win a race along with Brazil 2012. So while they momentarily rose to the top on some circuits (including Monza '09), they couldn't quite remain there, and dropped to midfield, where they have remained. And you could excuse Brazil '12 being wet, which suited their setup and Hulk.
Only other instance I can remember, when FI ever had a shot at the victory, was Perez in Canada 2014, when both Mercedes hit trouble.
While they had established themselves solidly in midfield, 2014 regulations and having the "right engine" helped them a lot. So they comfortably ended in top 6 in the constructors despite having some troubles with chassis performance in some parts of 2014 and 2015. But their chassis has gone ever stronger since then and they are still going well, despite Merc having gradually lost their PU advantage by now.
The funding obviously has remained a mystery for a while.
Despite having reportedly been "on the financial limit" for much of the decade, they managed to survive for quite long and kept designing good chassis. So whatever problems Mallya had, he made a good job of not letting the team getting distracted by it.