So my roommate and good friend of mine for many years has a crazy girlfriend that he wants to move out. She is not on the lease, she doesn't have any source of money so she doesn't pay any money towards anything here. The problem is he wants to be civil about this breakup (I personally don't think that possible with this girl but told him I would try). He's also military and she is beginning to affect his work which is very important to him. What are his options on getting her out "politely" through either civilian services or military services WITHOUT having anything on his military record?

Eviction is the only way to remove a tenant or roommate who won't leave voluntarily, even if they aren't paying rent. Laws go by state in the US, but in general someone is a tenant if they can prove that they reside there, through mail sent to the address, furnishings, etc. often after only 30 days.
So he can be 'polite' but it's never nice. He starts by giving her written notice that she has 30 days to move (he could also start charging rent and then lack of payment becomes part of the eviction). This should be by the first of the month if that is when he owes rent. That is followed by a formal eviction process done through his local courts.