Hello Sasha!
I am not sure the exact level of English needed to answer this question but from what I have seen you write in your questions I am sure that you could rise to this challenge!
It makes things easier to break it down and look at the meanings of each word and start from there. The word "exploits" in this context means bold and daring feats (the way he carried out the battles)
So let's look at each meaning:
vigorous exploits - vigorous meaning powerful, forceful and energetic
daring exploits - daring meaning courageous, bold and fearless
impulsive exploits - means without forethought or propper planning
violent exploits - meaning with physical force intended to hurt, damage, or kill.
So I would say "daring exploits" because the text seems to be painting him as a hero and not as "impulsive" (a warrior wouldn't go to battle on impulse), "violent" (wars are always violent anyway), or "vigorous" (this is not a word that I would use in this context.
I hope that helps!