Yes a felon can run for president and if they win the election and are voted in the felon can take office as president of the United States.
There are no restrictions against a felon running for the office of the President.
Only the Constitution may spell those restrictions and it doesn't address the issue.
This is because the Constitution is minimalist in nature; the belief was (and remains) that the individual states' would be sufficient to provide all the additional vetting of the candidates for the Office of the President of the United States.
Being a felon though may hurt your chances of winning the election for president because some people will check your background out and if you've done anything bad to be convicted as a felon then they may not want to vote for you.
However there was a felon that become a mayor and is a good person now so it's possible if you can actually show that you've changed your ways.