Hmm. Well it's certainly grammatical. The problems are twofold: first, it emphasizes the adverb. They didn't mind his killing the chickens, but it was doing it brutally that annoyed them. Perhaps you meant this. The normal order 'brutally killing' doesn't have this emphasis - he killed them, and it was brutal, but it was basically the killing that annoyed them.
Second, the sentence in writing approaches ambiguity. It's hard to see what 'brutally annoying' someone is, so we don't take it that way; we can see that 'brutally' must refer back to the killing, but it is a more natural place for an adverb modifying 'annoyed'. So we hesitate at it. In speech, there wouldn't be a problem - intonation would tell you which it was:
[His killing the chickens brutally] [annoyed everyone]. (natural meaning)
[His killing the chickens] [brutally annoyed everyone]. (less natural meaning)
You can't use 'he' here. The subject of a gerund-participle has to be 'his' or 'him', both equally good, and I don't think it makes any difference to the adverb placement. Whether 'his' or 'him', you get the same emphasis and the same ambiguity (if you used an adverb that could apply to both verbs).
Actually, I want to know whether the verb 'killing' is a verb or a noun, and whether the genitive pronoun 'his' is a real subject of the verb 'killing' or just a possessor as in 'my job/his name/your book'.
Some articles say that in such case the genitive pronoun is much like a possessor. However, it seems to me that such kind of pronoun should be a real subject as in 'He killed the chicken'.
So subject or possessor ?
Which one makes more sense to native speakers ?