You aren't really asking a legal question. Your felony conviction is a public record, but whether it shows up on a background check depends on how deep the company doing the background check digs into the records.
If you lie on a job application and they EVER find out about it, they can fire you on the spot, even years later.
It would help if you listed the exact statute under which you were convicted. I suspect it might be Penal Code 246.3, negligent discharge of a firearm, which can be reduced to a misdemeanor. Felonies that can be reduced to misdemeanors are known as "wobblers."
I am not as enthusiastic about so-called "expungements" as some other lawyers, but you can request that at the same time you move to reduce your felony conviction to a misdemeanor.
Under Penal Code §1203.4, a person who is granted probation can apply to have the case dismissed after probation is complete. This is sometimes called an expungement, but it doesn't really expunge anything or seal your record. The conviction is still a part of the court's public records, which will also show the case was dismissed. You still must disclose the conviction when applying to be a police officer or for certain other jobs, or if asked on an application for a state license or to run for public office. It will not remove any restrictions on your right to own a firearm or relieve you of the requirement to register as a sex offender. The conviction can still be considered a prior offense; for instance, a prior theft conviction could make any future petty theft a felony, or a prior DUI conviction could be used to increase the punishment in subsequent DUI convictions.
BEWARE of law firms that promise to "clear your record" and charge thousands of dollars. The process of filing a Penal Code §1203.4 petition is pretty simple, and you can do it yourself with forms available from the court clerk's office, especially if you completed probation without any problems. If you had a probation violation along the way, you may want an attorney's help, but the guys who charge thousands to file simple paperwork will probably overcharge you for that, too.
California Labor Code §432.7 says employers can't ask about any arrest that didn't result in a conviction, inquire about it from other sources or use it in a hiring decision.
Some attorneys interpret this Labor Code section to mean you don't have to disclose a conviction that was dismissed under Penal Code §1203.4, but I usually advise clients to disclose it, with an explanation that the conviction was subsequently dismissed.
Please understand that this is a general discussion of legal principles by a California lawyer and does not create an attorney/client relationship. It's impossible to give detailed, accurate advice based on a few sentences on a website (and you shouldn't provide too much specific information about your legal matter on a public forum like this site, anyway). You should always seek advice from an attorney licensed in your jurisdiction who can give you an informed opinion after reviewing all of the relevant information.