So many good replies already that it almost seems like a waste of cyberspace to add to this being that I don't believe I can say it any better than has already been said.
Do you ENJOY killing a deer?
No....I love to hunt.
"Why do you LIKE killing things?"
I don't.....I love to hunt.
The thrill of hunting like so many things in life, is in the chase, the journey, not the destination (kill). That would include but not be limited to:
An attempt to put pieces of a puzzle together correctly. A puzzle that requires at least some understanding of natures vague clues. A crisp track, a faint trail, a tuft of hair blowing in the wind on a barbed wire fence, a sapling with missing bark, a broken branch above some freshly scraped soil, or even a bloody antler laying on the thawing earth. All tell a story, and add to the experience, your journey.
To be a consistently successful hunter, it requires some familiarity with prey that survives and thrives by instincts and senses that would make even the infamous "Spiderman" envious. Time and logic mean nothing to this quarry. Those are qualities that apply to "our" world and usually hinder us on our journey.
Hunting is interacting with nature in such an intense way that you fear the beating of your heart will betray your position, yet your presence is no more noticed than the morning mist fading with the advancing sun light. Hunting is competing in an arena where you are always the visiting team, a huge underdog, and there are no time outs. It is the thrill of a few victories and the memories (not all bad) of many defeats. Though the game is played for keeps it is all part of a magnificent journey, in the end, for all the competitors.
The kill is really anticlamactic. It is the part of the hunt that is the least "enjoyed". We don't have a catch and release option when we speak of controlling numbers in the herd. Killing is a necessary evil for lack of a better word. But if you think about it, there can be no sustaining life unless something dies. That is the law of the land. Death is a foe we must all face. It is part of the "circle of life". Being that we are at the top of the food chain we play a significant role in that circle and not just with wild animals.