I've just started reading the Wolverine and X-Men series without reading all the backstory and was wondering how on earth it was these two that ended up heading the new school, especially Wolverine? I would have thought there were much better and more suitable options available amongst the X-Men.
The reason for the duo to be headmaster and mistress is that:
1. Kitty Pride represents respectability and a defensive face for the public.
2. The Wolverine represents a mutant whom all mutants respect for his power and prior acts.
This is discussed in the first couple issues of Wolverine and the X-Men prior to the state inspectors showing up.
After the events of X-Men: Schism the team split into two groups. I won't go into the ideologies that led to the split (as it's a fairly new storyline that I don't want to spoil), but Cyclops now leads a group of X-Men based in Utopia (an island off the California coast) while Wolverine leads a group at the School for Gifted Children back in New York.
There may be X-Men who are better suited as leaders, but Wolverine has historically settled himself into a role of mentor or older-brother to a number of younger members. Kitty Pryde and Jubilee were probably the best examples of this. Becoming a headmaster is something that actually seems right up his alley.
As I recall, Kitty is one of the few members who actually went off and studied at a more traditional (read: non-mutant) college. This also seems like a reasonable fit.