Pain is just an information
It is just your body saying that "Something bad is happening!". Yes, you can feel so much pain that you almost cannot function... but this is all in your brain. This means, that there are certain methods to learn how to deal with it - you can meditate or... you can get angry.
Adrenaline tells your body how to reallocate resources, causing the physical responses, one of which includes the release of endorphins, neurotransmitters that act as your body’s natural painkillers. With endorphin release, your after-accident pain may be partially or completely masked. source
I'm not sure how good Logan is on the meditation techniques, but he definitely knows how to get angry, to the point where he is sometimes dangerous to his friends.
Wolverine has had a temper and bloodlust ever since he was introduced, with his fighting spirit and indomitable will part of what allowed him to take on the Hulk in his very first appearance. This penchant for violence is showcased especially whenever he has something to lose or he has already lost something important. For instance, seeing his teammates in mortal danger will set him off, endangering not only his enemies but anyone around him. source
Nice example here:
He is lost in some sort of berserker rage and doesn't feel pain anymore
Now, since this has been downwoted due to lack of of the movies, let me bring some examples from the movies themselves:
When Wolverine is wounded, he can definitely feel pain (at least judging by his facial expression):
In the first X-men movie, Magneto immobilises Logan in a such way, that the claws are facing into his own arms. You can see that Wolverine makes a few deep breaths, preparing himself mentally before slicing his own body to free himself. He does feel pain, but he is prepared to take it - this would be the "meditate through pain" method.
In contrast, you can see the second way (rage) for example in the Wolverine movie, where he is pierced by multiple arrows:
His face is just a tranquil fury