The answer is simple: because SHIELD is an international policing force that is in the way of their ultimate goal.
That alone is sufficient motivation for any "bad guy" to have a beef with SHIELD.
So, since all criminals are, in fact, criminals, and will necessarily run afoul of the law at some point in their career, you're saying it would be a good strategy for them to go out and start killing cops and attacking police stations?
Sorry: crime doesn't work that way. Criminals try to operate in the shadows and *avoid* the law as much as humanly possible. In fact, that's exactly what Killian did in creating the Mandarin smokescreen to divert the authorities and military from investigating the true cause of all those blasts.
Why would he get involved with an organization like Centipede?
I'll answer your question with another question: Why would he get involved with an organization like The Ten Rings? At least Centipede seems to have more advanced weaponry and organization than The Ten Rings showed in IM1... Who's to say Centipede isn't what's left over of 10 Rings, with a rebranding?
Killian was only using Ten Rings as, stated above, a smokescreen. By all appearances, in 2008, Ten Rings was an ancient terrorist organization that had long since become a shadow of its former self and was now a loose assembly of disjointed cells with pissant warlords like Raza commanding them. Think: the Assassins (the real organization; not the one in the video games). Think: al-Qaeda.
Stane employed Ten Rings for one simple task: to kill Tony Stark. Raza ****ed
that up royally and indirectly ushered in the Heroic Age. Rather than kill Tony, he made him into a genuine superhero, who came back and essentially destroyed what was left of Ten Rings. Killian then seized the opportunity to co-opt the Ten Rings name to cover up his Extremis "accidents." He also co-opted an ancient Ten Rings leader figure, The Mandarin, and placed an actor in that role.
It's not that hard to map out. Ten Rings were just pawns of Stane and Killian. Who may or may not even have a connection.
Unless the "Hail to the King" one-shot throws a monkey wrench in everything that Drew Pearce, Shane Black and Kevin Feige said about IM3, then yes, Killian is officially The Mandarin. All three of those gentlemen specifically said so in interviews.
And considering that Drew Pearce also wrote the "Hail to the King" script, I wouldn't bet a single dirty penny on him changing his mind and retconning Killian's story for the sake of a few angry fans who didn't understand the Mandarin Twist at all.