The Noda B'Yehuda wrote a lengthy teshuva (Yoreh Deah, Teshuvah 10) addressing hunting.
1) Perhaps hunting is a violation of "צער בּעלי חיים" (causing suffering to creatures) or "בּעל תשׁחית" (unnecessary destruction)?
A: This would not be considered "causing suffering to creatures."
since that prohibition only refers to causing suffering to an animal while it is alive.
This would not be considered "unnecessary destruction"
a) This prohibition only applies to property that belongs to someone. In this case, we are talking about wild animals.
b) Furthermore, unnecessary destruction only applies only when there is no benefit as a result of the destruction. In this case, the hunter may use the skins, and so the act does not fall into the category of unnecessary destruction.
2) Nonetheless, the Noda B'Yehudah writes:
ואמנם מאד אני תמה על גוף הדבר ולא מצינו איש ציד רק בנמרוד ובעשו ואין זה דרכי בני אברהם יצחק ויעקב
- However, I am surprised by the matter itself. We find no hunters other than Nimrod and Esau, and this is not the way of the children of Avraham, Yitzchak, and Yaakov.
3) Hunting in order to make a living vs Hunting for entertainment
Hunting for livelihood if someone has a need for this and makes his living from it, then there is no issue of cruelty involved .
Hunting for entertainment but if his main intention is not to make a living, then it is cruelty. Hunting for entertainment or pleasure is a form of cruelty.
4) Hunting puts one in danger
ועכשיו אני אומר אפילו איסורא איכא שהרי כל העוסקים בזה צריכין להכנס ביערות ולהכניס עצמם בסכנות גדולות במקום גדודי חיות ורחמנא אמר ונשמרתם מאוד לנפשותיכם … ומעתה איך יכניס עצמו איש יהודי למקום גדודי חיות רעות
Now I say that it is even forbidden, for anyone who engages in this must enter the forests and place themselves in great danger, in places of packs of wild animals And the Merciful One said: “Take great care of yourselves” (Deut. 4:15).... how can a Jewish man insert himself into a place of packs of wild and vicious beasts?
The Noda B'Yehuda concludes, writing:
ולכן השומע לי ישכון בטח השקט ושאנן בביתו ולא יאבד זמנו בדברים כאלה
One who heeds me will therefore dwell safely, tranquilly, and contentedly at home and not waste his time with such things.
BOTTOM LINE: unless it's for one's livelihood, the Noda B'Yehuda rules that hunting is forbidden.