What kind of person has their self-esteem tied into whether a d20 roll was sufficient to succeed at a climb check in a roleplaying game?
Abuse survivors, or people with clinical depression or anxiety can behave in in such manners. It isn't that their self esteem is specifically tied to the game - they will tend to have similar reactions to many things in life, especially in social situations. Small frustrations get blown out of proportion.
If the person really has such a tissue-paper self-esteem as you suggest, then he should be seeing a therapist rather than spending time playing make-believe.
Maybe he is. But if he is, he doesn't get to sit in the therapist's office 24/7. He's got to have a life, too, and friends - very few troubled people get better without a support structure.
The guy sounds like a whiner and a passive-aggresssive roleplayer and a drag to be around. Give him the boot. Unless you are a therapist doing pro-bono work, and the other players don't mind wasting their time with a whiner.
If you don't have friends with serious troubles like this that needs your support, count yourself lucky. They aren't uncommon, especially in the geekish communities.
I think it admirable that you want to help, airwalkrr. Friends try to help each other out, and that's cool.
I don't know what your table dynamic is like, but maybe taking the discussion out of the game for a bit (maybe just GM to player) might help. Make it non-accusatory, and about helping him, not about how he's a burden on you. Along the lines of, "Dude, we like you, you're an asset to our game. You seem to be down on yourself and your character a lot - we *like* your contributions, and we don't see the problems you see. How can we help to get you around feeling like that, so you can enjoy the game more?"
Just the explicit statement that you accept him and want to help him have more fun can mean a lot to someone.
There's not a lot beyond that we could say, not being mental health professionals, and not knowing the person in question.