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I am sorry to hear about this situation. On this website, I do not always get to give good news, and this is one of these times.
The short answer is: it is virtually impossible to do unless the Plaintiff agrees to cooperate.
The Defendant would have to file a motion to seal, asking the court to seal the record. The Court has discretion in doing so going off a discretionary set of subjective factors dealing with public information versus rights to privacy (see HERE) but suffice it to say, eviction records are generally NOT SEALED as matter of due course. The only way to get this done is to have the Plaintiff agree to sign off on the motion, and even then the Judge may still deny the request.
Assuming the above is successful, this should have the matter blocked from a general background search, but it would still show up on credit checks (which are often part of background searches). To get this off a credit bureau's history, one would have to CHALLENGE the history entry with each of the three major credit bureaus. Assuming the Plaintiff does not respond to the challenge within 30 days (again, this is why their approval is needed) and the court record is blocked, the bureau should then void the entry.
As you can see, this requires cooperation from the Plaintiff in the suit, then. Please note: If I tell you simply what you wish to hear, this would be unfair to you. I need to be honest with you and sometimes this means providing information that is not optimal. Negative ratings are reserved for experts who are rude or for erroneous information. Please rate me on the quality of my information; do not punish me for my honesty.
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