How should one hunt animals?

I think in some ways this requires answering the question from different points of view.

In terms of the doctrines of Christianity itself, there is no ban on meat consumption. God isn't going to send you to hell for eating meat and all plants and animals are considering clean. So in that instance, there's nothing wrong with hunting for sustenance.

However, if you dig deeper into the lives of the early church fathers some of them reject eating meat because they believe it inflamed the passions. I'm sure this is what led to the influence of limiting/abstaining from meat in monasticism today. One of the many reasons I choose to remain vegetarian is from this St. Issac quote:

What is a merciful heart? It is a heart on fire for the whole of creation, for humanity, for the birds, for the animals, for demons, and for all that exists. By the recollection of them the eyes of a merciful person pour forth tears in abundance. By the strong and vehement mercy that grips such a person’s heart, and by such great compassion, the heart is humbled and one cannot bear to hear or to see any injury or slight sorrow in any in creation. For this reason, such a person offers up tearful prayer continually even for irrational beasts, for the enemies of truth, and for those who harm her or him, that they be protected and receive mercy. And in the like manner such a person prays for the family of reptiles because of the great compassion that burns without measure in a heart that is in the likeness of God.

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Merciful Heart: St. Isaac the Syrian | Orthodox Quotes

I like this blog as it shows monks caring for animals:
Orthodoxy and Animals « ORTHODOX CHRISTIAN FAITH AND LIFE

Today, there's a lot more info on vegetarian and vegan options, so it's easier to be one today than it used to be 20 years ago.