There are two ways to order Wolverine's appearances.
Wolverine is forced to kill Jean Grey/Dark Phoenix.
- X-Men Origins: Wolverine (2009) - This shows Wolverine's discovery of his powers, working with Stryker, and the bonding of adamantium to his bones/claws.
- X-Men: First Class (2011) - only contains a quick cameo of Wolverine. The conversation between he, Xavier and Magneto is briefly referenced in Days of Future Past
- The Wolverine (2013) - Shows Wolverine during the end of World War 2, and after X-Men: The Last Stand.
It specifically references his killing of Jean. This is where Wolverine loses some of his adamantium, specifically in his claws. The post-credits scene leads into Days of Future Past.
- X-Men: Days of Future Past (2014) - Wolverine time-travels from 2023 back to the 1970's. This contains brief appearances of Stryker and references the conversation from X-Men: First Class.
The "future" scenes take place after The Wolverine, but Wolverine has adamantium claws again. The ending shows that events of X-Men: The Last Stand have been undone.
he escapes, and inadvertently saves the young X-Men in the process
- Logan (2017) - takes place in 2029, dealing with the aging Logan and Charles Xavier as they come across a new young mutant. It is heavily implied that
all of the other X-Men are dead.
In-Universe Chronological order
Since there are time jumps here, there is no 100% clean way to watch them in a purely chronological order, but this order would cause the least confusion.
- X-Men: First Class (you could technically omit this entirely if you're only concerned with Wolverine)
- X-Men Origins: Wolverine
- X-Men: Apocalypse*
- X-2: X-Men United
- X-Men: The Last Stand
- The Wolverine
- X-Men: Days of Future Past
*these films exist in the alternate timeline created by the events of X-Men: Days of Future Past.
You could also swap X-Men: First Class and X-Men Origins: Wolverine and his cameo might arguably make a little more sense, but you wouldn't really gain or lose any insight regardless of which of those two you watched first.
Since DoFP comes both first (excluding First Class) and last, I could see the argument being made for watching it first. But since the net result of DoFP is the resetting/modification of the main Fox timeline, I think it still makes more sense to watch it last. This way you understand the gravity of the final scene at the school.
So which order should you watch them in?
Honestly, I'd go with the release order. Watching them in chronological order really doesn't offer any additional insight or explain anything any better, and may actually confuse things more since it would weave two separate timelines together.