An FBI background check is nation wide, so the fact that it occurred in another state is not going to conceal it. Sometimes it is best to just present the facts to the recruiter and let them suggest what might be needed.
You can get a copy of your FBI background check. See the following link: http://www.fbi.gov/about-us/cjis/nics/general-information/cgbrochure . Good luck.
Yes the background check will show your case. I don't believe it will impact joining the military but as mentioned you should do your own background check first to know what you need to do and say.