If I plead guilty for first offense felony will it show up on my background check?

Yes, this conviction will appear on background checks. You must wait 5 years before you can petition the court to have it expunged off of your record. Each employer will handle the conviction differently. Some employers may choose not to hire you based on the conviction, others may still give you a chance. You should be prepared in interviews to address this issue and explain it in a positive way. i.e.

If your felony happened in Missouri, there are only a few types of felonies that qualify for expungement after 20 years of good behavior. Expungement is only possible if you were actually convicted. It sounds like you have not been convicted and won't be if you successfully complete probation without any violations. If you were arrested, then the record of the arrest will be on your criminal background report and it can not be expunged. Only arrests and convictions are supposed to appear on criminal background reports, but sometimes the record of the probation also shows up.

You will not be able to expunge the felony from your record unless you are pardoned or if your conviction is overturned. If you are on a deferred probation then it might be possible for you to ask for a non-disclosure 5 years after you successfully complete the deferred probation.