Tthis is kinda vague, but I read a couple days back a thread that had a list of 4 charges from 2013 against Brown, including one firearms specification but did not save it. I only mention it as I also read info that pertained to those charges has vanished from the internet. Go figure.
From what I read that was a different person with the same name that lived something like 60 miles away.
As far as can be determined the Mike Brown in Ferguson had no criminal record of any kind.
It turned out that those charges were against a different Mike Brown.
Brown was only 18.
If he has a long record, it would be a juvenile one, and is likely sealed.