Position applied for: Investigative Team would probably fit him best, though dispatch may be a close second.
Is your character psychologically fit? Please elaborate: Definitely for something like this. He would be very interested trying to figure out the prison system for the benefit of everyone and should there ever be a reason to do so, he is more than happy to risk his life to save someone. For the most part he can be considered sane, though he's very animated and a tad hyper, and should be feel inclined to he'll wander off to do some work on his own. Authority might be a bit of a problem for him, as he's used to being treated as a higher up ( like in U.N.I.T) or considered the expert. Also he ran from his people because of their oppressive society and rules. Even if asked, there is no way he would kill anyone for any reason, though he has no qualms about not offering second chances.
How much of this would the interview team be able to read?: The Doctor is an extremely skilled in telepathy, so a profiler with telepathic ability may not be able to read his mind much, or at all. That'll be case by case. Otherwise it may be quite easy to read his emotions as he tends to be very expressive, although he'll hide his feelings and try to distract the profiler if there's something he really doesn't want to be known.
What skills/knowledge/powers/experience does your character possess which would be an asset to the position(s) they are applying for? He's an incredibly skilled telepathic, though he is against invading another mind and won't attempt it without permission. Then there's his vast intelligence and knowledge of languages, cultures, mathematics, mechanics, time travel, etc. It would be safe to say he's a know it all. If there's anything he can cobble together from different technologies, he'll do it. Also he's a fairly decent sword fighter and is skilled in numerous martial arts (though he never uses them). Under no circumstances will he shoot a gun, though it's very possible he's got great aim.
How competent is your character in his/her skill set?: He is very competent in all his fields, usually being touted as the expert on all things alien on Earth (and other various planets). However, if something goes horribly wrong and everyone is in horrible danger, there is a very high chance he'll become very emotional and very vengeful, so this may affect his thinking. This version of the Doctor has an even higher chance of running that risk, as he is mister blood, anger, and revenge.
Would you like to play out the interview?: N
Would you like to play out the skills test?: Y, but only if it is convenient for the other members. A psychic test would be entertaining.