Known only as Logan to his very few friends, Wolverine is a distant man with a past shrouded in mystery. His memories reduced to a flashing series of contradictory images, Logan comes to the Xavier Institute at the behest of Professor X. While taking the time to regroup and sort himself out, he soon finds himself assuming the role of teacher at the Xavier Institute, teaching the kids various martial-arts and meditation techniques.
Wolverine will remain our resident enigma. The object of fear, awe and speculation on the part of the students, he will inspire the X-Men in good ways (providing battle tips after clandestinely watching Danger Room sessions) and bad (inspiring conflict between our core group of X-Men simply by virtue of his loner status and rebellious ways).
Wolverine will not be part of the team. However, he will join them for an adventure or two, and maybe hang back in the shadows to keep our kids from catastrophe (or ambush). He will not be there to pull their fat out of the fire or to stop them from being heroes - it's their show! Instead, he will be there as a presence, providing an ethos, the voice of rebelliousness ever creating friction on our team (whether he means to or not).
Knowing the evil in people's hearts (from battling such enemies as Sabretooth), Logan knows that mutants must be proactive in terms of dealing with threats, otherwise normal folks will think of "people like us" as needing to be rounded up. In "The World According to Logan", the key isn't blending in, as Xavier advocates. It's stopping the people who refuse to blend in from becoming a problem to normal people and, by extension, other mutants.
Wolverine's mutant powers are his hyper-keen senses, enabling him to see farther, and in almost total darkness. He has the tracking ability of a bloodhound, and a nose just as sensitive. He can literally sense danger. He also possesses a healing factor, enabling him to survive any wound. He isn't invulnerable - he wouldn't walk away from a downed 747 - but no matter how badly he was hurt by the crash, within a week he'd be right as rain. Oh, and by the way, he also has these huge Adamantium claws that pop out from the back of his hands. Who put them there? Well, that's all part of Logan's missing memories, and what little he does remember about them tends to contradict everything else.
Wolverine is a scrapper; he's ornery when cornered, and loves nothing more than a good fight. But he realizes that these kids have to learn to use their powers and save the world, so he doesn't interfere with their education.