Wolverine mini-series by Chris Claremont and Frank Miller- Wolverine goes to Japan only to find out that the woman that he loves, Mariko, is engaged to an abusive man and that her father has ties to the criminal underworld. Then he gets attacked by a bunch of ninjas. This is the definitive Wolverine story.
Kitty Pryde and Wolverine by Chris Claremont and Al Milgrom- This mini-series is about Kitty Pryde going to Japan to see what wrong her father has gotten into. Along the way she gets taken over by Ogun the devil himself and Wolverine must help Kitty Pryde over come the devil that has taken over her.
Wolverine: Weapon X by Barry Windsor-Smith- This story details what happened during the Weapon X experiments that gave Wolverine his adamantium skeleton.
Wolverine Origin by Paul Jenkins and Adam Kubert- After nearly 30 years, Marvel decided to finally reveal Wolverine's past. This details where he grow up, his real name, how became known as Logan, and how he became the man we know him as today.
Wolverine: Enemy of the State by Mark Millar and John Romita, Jr.- Wolverine has been brainwashed by the combined forces of Hydra and the Hand and is going after the heroes of the Marvel Universe.
Wolverine: Old Man Logan by Mark Millar and Steve McNiven- A story set in the future where Logan's a pacifist farmer in a country being run by supervillains. He reluctantly agrees to help Hawkeye in one last adventure.
Get Mystique! by Jason Aaron-Classic rivalry between Wolverine and Mystique. It also has quite a few flashback scenes, covering the past between the two.
Uncanny X-Force by Rick Remender and various artists- Wolverine leads a covert X-Force group whose mission is to kill any target they deem too dangerous to live. Their first target: Apocalypse.
Logan by Brian K. Vaughn. 3 issue mini-series