In the opening credits, The Wolverine hits a soldier with a weapon; blood and a bit of his windpipe is seen very briefly.
Wolverine rams his claws through Victor's hand with his left hand and you see the claws go through the back of his hand. Victor screams in pain, then Wolverine shoves his claws into Victor's chest with his right hand causing Victor to momentarily stop screaming in shock.
On several occasions, Wolverine kills people by fatally wounding them with his claws.
William Stryker tries to kill Wolverine by shooting him with special bullets, we see him get shot several times in the chest, and then repeatedly in the head at close range. Minimal blood is shown and we see fragments of his metal skeleton exposed through his skin.
After Wolverine emerges from the tank he had the adamantium injected into him with, he is shot several times by Zero. No blood is shown.
The character, Zero, is a marksman and, as such, shoots multiple people by dual-wielding two pistols. In a later scene, he shoots an elderly couple, but we do not see any blood.
Victor Creed stamps on Wolverine's bone claws and breaks them. Wolverine howls in pain.
A female character is killed and blood is seen all over her T-shirt.
We see the bloody severed head of a real wolverine.