If my father who is a US citizen wants to file an I-130 so I can become a resident, can I file the petition and still stay in the US?

More information is needed in order to provide an immigration legal analysis. Depending upon the circumstances you already may be an American citizen by virtue of being the child of a U.S. citizen. If those circumstances do not apply to you, your parent may be able to sponsor you in the Family-Sponsored Third Preference visa category (married sons and daughters of U.S. citizens), but for that you may need to be in valid nonimmigrant status at the time of filing for adjustment of status. If you were to apply in that category through consular processing abroad, note that the duration of any unlawful presence in the U.S. may subject you to a 3-year or 10-year bar to re-entering the U.S. There really is no substitute for consulting with an immigration attorney who, after learning all of the relevant information about you and your family, could advise about immigration eligibilities, options and strategies, and who then could offer legal representation in the often quite complex application process.