Well, the only two qualifications for the job are being at least 35 years old and being a natural born US citizen, so lots of people qualify.
Outwardly, it takes a force of will, and a certain amount of hubris. Being a well known person, because of other accomplishments, schooling, military training and leadership, former public office, don't hurt your chances either.
Not having a bunch of skeletons to keep buried doesn't hurt. The ability to make good public speeches, and how likable of a person you are.
But none of that is a hard fast rule, sometimes you just need to be the right person, in the right place, at the right time to catch lighting in a bottle.
It is said that we all get our 15 minutes of fame, but can you, and will you, seize on the opportunity to use that to propel yourself to public service, and will fortune smile on you.
There were some Presidents that were clearly bred for the job, like JFK, but it still required a great deal of just pure luck, not getting killed in war, and being the right person at the right time for the job.
In no small part, does it take ambition, if you are just going to fold the first time things get hard, or even the 10th or 20th time, you are not going to make it.
Being a former criminal won't help you, but it's not something that can't be overcome by ambition and force of will.
Being high born doesn't hurt, but there have been plenty that were not.
Mainly, it's not an opportunity that's just going to fall in your lap, force of will, ambition, and determination, with no small part of luck.