How do we know that Thorin Oakenshield is... well, Thorin Oakenshield? Because he tells us so, of course, and because the other dwarves tell us so. So does Gandalf, and for that matter Dain Ironfoot. Except... we have no hints that Gandalf knows him for any reason other than that Thorin has told him who he is. As for Dain, he was barely more than a child when they fought together outside Moria - can we really trust him to recognise if 'Thorin' isn't actually Thorin?
So that leaves the Company - those dwarves who are travelling with 'Thorin' in hope of material gain. Let's imagine for a moment that Thorin (son of Thrain son of Thror) survived the Battle of Azanulbizar, and indeed survived to go with Thrain, Balin, and Dwalin on their expedition to the Lonely Mountain. And then, let us imagine he was captured at the same time as his father - and killed.
That would have left the dwarves of the Blue Mountains with no king of Durin's line - the title would have fallen on Dain, far away in the Iron Hills. Remember, Balin and Dwalin's company had set out for the Lonely Mountain with those two dwarves who would be able to claim its riches for the Blue Mountains. Now, even if they somehow, someday retook the Mountain, the wealth would pass to Dain. They would see very little.
And so they made a pact. One of their number would be injured, in such a way that his face could be bandaged. He would be 'Thorin' henceforth - the son of the king, a prince who hadn't been particularly prominent in the people's minds while Thrain lived. He would, of course, have to be a traveller - spending too much time in the halls would allow people an opportunity to figure out something was wrong. And the loss of his father was a perfect excuse for that.
Fast-forward a hundred years, and 'Thorin' has settled nicely into his role. He and the other pact-dwarves begin to seriously consider making another attempt on the Lonely Mountain. They create a Company, consisting of the pact-dwarves, and others who never knew Thorin before his father's disappearance. They also bring along Fili and Kili, as their 'proof' of 'Thorin's' identity - they weren't even born when the real Thorin died, so they never knew any other face.
Would Dain have seen through the deception? It's hard to say - and he never got the chance. As it turned out, 'Thorin' died in the Battle of the Five Armies - quite possibly disfigured for real this time. As Balin and Dwalin had feared, the rule passed to Dain - but they didn't do too badly out of it, in the end.