What are the chances of a citizenship application for someone with felony conviction?

Generally, a felony conviction is automatic grounds for permanent bar to US citizenship.However, the crime of conviction must meet the federal definition of a felony. What constitutes a felony according to state law is not necessarily so defined under federal immigration law purposes and vice versa. Also, due to the ever expanding pool of the types of crimes that Congress added to the felony definition, your own crime of conviction may or may not be a felony as defined under federal law. Also, for your crime of conviction to be a felony, that permanently bars US citizenship, it must have been so classified at the time of your conviction. You will need to consult with an experienced immigration attorney.

For naturalization it will depend on how old the conviction is and whether it reflects on their current moral character. With a felony, I would suggest at least consulting with a lawyer if not having a lawyer handle the process.