How do I go about the process to petition my mother to be a legal citizen?

Significantly more information is needed in order to provide a legal analysis of ways you may be able to sponsor your mother for becoming a Lawful Permanent Resident (and five years after succeeding with that, she could petition to become a citizen). This includes, for example, information about your own immigration status (Are you a U.S. citizen? Are you a Permanent Resident? Do you have some other status?); information about your age; information about your mother (How and when did she enter the U.S.? What is her current immigration status? What immigration-related applications every before have been filed by or for her?); Etc. There really is no substitute for you and your mother to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.