If Wolverine could get past the initial application process and hide the fact that he’s an alcoholic, the recruiter would then send him to MEPS (Military Entrance Processing Station) for a physical evaluation. At this point, they would probably discover that he has an Adamantium skeletal structure through x-rays.
Now, it could go a couple of ways at this point. First, they might reject him outright for not having ‘normal’ characteristics of an average human being. (How many people do you know, personally, that have an Adamantium skeletal structure??) Secondly, and more plausibly considering our government and their secretive, manipulative ways, they’d let Wolverine enlist and after (or maybe even before Boot Camp), they’d send him off to some secret military or government facility to study him and exploit him and use him to whatever extent might fit the government’s already unethical and immoral purposes.
An alternate theory too is that (after seeing what Wolverine is capable of in the X-Men movies), he might not tolerate any shenanigans perpetrated by the government and end up kicking the living shit out of any scientists or doctors or military brass who attempt to do any tests or experiments on him in the first place.