Alongside Dumbledore's achievements, as listed by rand al'thor, one might also mention the personal traits and abilities that preconditioned them. On the whole, I believe, these are unrelated to magic essentially, so it is easy to fathom that Dumbledore would have become great in something else had he not been a wizard in the first place.
Yes, Harry, blessed as I am with extraordinary brainpower, I understood everything you told me. I think you might even consider the possibility that I understood more than you did.
[Harry Potter and the Half-Blood Prince]
Brainpower can be pretty much anything, from problem solving to strategic thinking to empathy, but Dumbledore clearly has a lot of all. It obviously eased things for him from early on, because of the very positive feedback effect on further learning.
In particular, Dumbledore can read people and situations. He saw through so many characters and events: Voldemort, Lockhart, Fudge, Slughorn, etc.
Also, Dumbledore is quick on the uptake, to borrow Rowling's wording (about Kingsley). This is difficult to emulate if not born with, and potentially crucial in situations where being a smartass does not really help (such as, on the roof towards the end of HBP).
Also, he is quite unconventional. Being an oddball in a few respects (such as, his family condition or his sexuality) made it easier for him, later, to be an oddball in general. People will talk about his passion for candy but I think he made his mark by telling his pupils openly about Voldemort's return, addressing them as adults rather than children.
Ultimately, of course, no attempt at such an account can be complete (especially after being repeated ad nauseam in the books) without mentioning
that he recognized love as the power that makes the world go round, so to speak, and also, he chose to explore it as a magic force. This was crucial in protecting Harry during his childhood years.