According to Felony Guide, under certain conditions, a felony can be expunged, or removed, from a person's criminal record. However, this will typical only apply at the state level. For federal background checks, a person's full criminal record check will still be taken into account.
As ClearUpMyRecord.com explains, expunging a felony charge mostly just requires filling out a lot of legal paperwork. The official qualifications for expungement vary from state to state, but typically require a period of probation where no other crimes have been committed, only a single or small amount of offenses, a lower level of severity to the offense, and that the crime does not fit into certain categories (such as rape, child pornography and other sex crimes and crimes against minors).