At first I thought this just referred to how Wolverine's heart gauge replenished itself over time, but in actuality:
You need to let yourself be killed. Instead of disappearing and losing studs, you turn into a skeleton for a short time before turning back into Wolverine, at which point the achievement should pop.
Like VGM 007 said - You first gain control of Wolverine in Chapter 4, and you can get this immediately after.
From ScuzzyBunny - You can put in a code to unlock a version of Wolverine, but this minifigure will not net you the achievement. So be patient and wait for Chapter 4, it's really not that long.