They did it by spending million to construct a lifelike prosthetic Hugh Jackman mask for the actor to wear. Then they threw it away because that's actually Hugh Jackman.
It's a Catch 22 because Wolverine telling the movie to go fuck itself makes the character compelling, and yet it is the very thing which keeps Wolverine from being in the movie. Wolverine is therefore prevented from being awesome and present in the same movie.