"What you need to win" is skewed, because two teams make up 5 of the last 6 cups. The game has changed significantly over that time, so I'll just look at those 6 years.
Great goaltending. Yes, having a great goalie can win you a cup. Look at Quick on the 2012 Kings, .94 SV% in the playoffs. Without him, they wouldn't have won that cup. BUT, no one was predicting he'd have that kind of performance. Goaltending is a high variance position. Having a great goaltender sometimes falls apart in the playoffs and having an ok goaltender sometimes works out. You can't plan for it. The best you can do is build like the Hawks so that even if your goaltending is just decent, the team is still good enough to win.
Centers. Blackhawks and Kings both had clear #1Cs. Bruins had a 1A/1B scenario. Blackhawks haven't had great center depth. Conclusion: good centers help your team win, get as many as possible. No secrets here.
Defense. Kings and Bruins had crazy defensive depth. Hawks have poor depth but Keith, who can play as much as two lesser defensemen. Have a very good defenseman or many good defensemen helps win
Gamebreakers: Hawks had Kane, Bruins and Kings played a consistent "Team First" game. Either works.
Overall conclusion: Get as many good players together as you can, play a good system, don't have any major areas of weakness, and don't depend on your goaltending.