I would like to know the same thing, it seems everytime someone asks this question, there is no serious attempt to answer it or it's just acknowledged but no ideas on what to do given.
That's because, IMO, there is no quick, easy, single answer.
Just look at the process from the start. First, you have to find a candidate that has pretty much a squeaky clean record. And who hasn't misspoke, done something wrong, made a serious mistake, etc.? Hell, in this day and age, we've got the media putting up stories about swatting a fly. WTF?
So, even the smallest of errors become magnified to absurdity with a media hungry for the next big story.
IMO, that's the single biggest obstacle to getting really good candidates.
I guess Hugo, as a starting point, I would say that we need to reform the privacy rights of candidates. I don't think elementary grades, high school grades, and probably not even college grades for a presidential candidate are really germane to making an informed voting decision. Nor do I think that a candidates medical history is anybody's business. You have to be at least 35, so by then, a candidate should have established himself/herself enough, publicly, for people to make an informed decision about their qualifications. And if they haven't established themselves, then I think that's a fair factor in deciding a persons vote.
This, IMO, is where we should start.