Legally, this highly depends on what country you are referring to and/or living in because of the legal restrictions. But most countries do not have any restrictions on this matter about "convicted" citizens running for the presidency.
As far as I know, there are only at least 4 countries that prohibits and immediately disqualifies convicted citizens that are running for the presidency: Afghanistan, Azerbaijan, Bangladesh and Turkey.
The problem mostly lies on how they will manage to rebrand their image to a positive one for the general public. Politics is dirty and a dirt likes this is very hard to clean but with how professional the media now at manipulating the pubic's opinion, it is not at all impossible for a convicted citizen to win the presidency XD
In my country, a convicted citizen cannot hold any political office. Once you're an ex-convict, every political dream you have would be blown away by the wind. So must times, opposition parties do find out about the criminal record of their opponents; mostly very strong opponents so as to discredit them before the election.
So the question of whether a convicted citizen can hold a political position is dependent on the country.
Depends entirely of the conviction, the circumstances, the whole background. There is no definite yes or no.