will this show up on the background check? Is this consider a conviction?

Right now it's going to show that you have an active arrest warrant and that you have an open and pending case for the public intoxication case. Signing the paperwork for deferred disposition does not end the case. Basically, the case is put on pause while you complete the terms and conditions of the deal. If you don't complete the terms and conditions, the judge issues a summons for you to appear in court and show casue why you shouldn't be adudicated and found guilty. If you don't respond to that a warrant will issue for your arrest. It sounds like that is the posture of your case. You need to hire an attorney to get that worked out for you. You'll have to pay a bond fee so that the warrant can be lifted. Your attorney can direct you to a bonsman or you can find one on your own. Once the warrant is lifted the court will place you on the docket. Once you're on the docket your attorney can go to court and speak to the DA about the case. At that point they'll either work out another deferred (if the judge will allow it at this point) or a conviction.  As an aside, if you have successfully completed the deferred, you would not have a conviction.