In a nutshell... Logan volunteers for Weapon X program after Sabretooth (who is killing mutants for Stryker) apparently kills his lover Silver Fox. He wants Stryker to give him the adamantium so he is able to kill Sabretooth. He then goes off, trying to find him, ends up in Vegas, including a nightclub full of mutants and a fight with Blob, then ends up back at the Weapon X facility, where he discovers Stryker creating an army of 10,000 Sabretooth clones (earlier draft had this as 10,000 Sentinels) and Silver Fox still alive (her death was all staged to get Wolverine to volunteer for the program..Logan had false memories implanted). Logan stops the Sabre-clones and Stryker shoots Silver Fox. Logan takes name Wolverine from an Inuit myth told to him by Silver Fox.
It has logic gaps --- Silver Fox's power is mind-control so why doesn't she mentally force Logan to join Weapon X, rather than planting fake memories of her death? And why is all the Vegas stuff needed - it sounds cool but odd. A lot of people were unsure about the 10,000 Sabretooth clones... In an original draft, Weapon X needed some mutant DNA samples (from Wolverine) to program Sentinels for mutant detection.... but why was only his DNA acceptable if many other mutants were known (and being killed!)?
So, I guess the script has potential but needs picking apart and beating into shape to iron out plotholes.