Wolverine, like the other two X-Men, has 25 available mutations (five levels in five categories). When you choose the difficulty level of a mission, it will show you the number of mutations that you will earn upon completing the mission. (Typically, higher difficulties yield more mutations).
You will only earn Wolverine mutations in missions in which you play as Wolverine. Note that if you replay missions, any mutations that you've already upgraded will remain in effect on your character, making harder difficulty levels easier to complete if you return to them later. I recommend doing so, as it will save you a lot of frustration rather than going for Superhero on the first try.
Some missions will yield the same number of mutations for multiple difficulty levels (i.e. Hero and Superhero both reward you with three mutations on a given mission). In these cases, I recommend that you pick the lower difficulty, as there is no need to complete the harder one in order to fully complete the game. But you will need to play the higher difficulties if the difficulty you beat a mission on did not reward you with the full mutation payout for that mission. I also recommend going for the Sentinel Tech and Weapon X Files on the lower difficulties, so you can concentrate solely on beating your enemies on the harder ones.
When upgrading Wolverine, I also recommend increasing your health mutations first, followed by (or closely balanced with) your attack damage, and the rate at which your Fury Mode depletes. Of course, If you're finding yourself healing often, you may want to sink a few points into that one as well.