How strong/vulgar is using "to bitch" in the meaning of "to complain"?
Solution 1:
I am a native New Yorker (from Harlem) living in Brazil, and am having some difficulty explaining to my English students the subtle mutations through which English word meanings pass. Because of the pervasive nature and influence of hip hop language in modern American youth culture, "Bitch" as meaning a vulgar woman, a whore, a "loose" woman, etc, has been replaced by the word "Hoe". Bitch has been relegated to the meaning of a person who is difficult to get along with, in the sense that they complain incessantly. Of course the context in which it is used gives the intended meaning, but for the most part the long-standing verbal meaning has also become the nominal meaning. "She's a bitch" now means that she "bitches" too much, or complains too much. In it's usage to refer to males, it means that he is a coward. I am now waiting for online dictionaries (except for the Urban Dictionary which already reflects this) to conform to this reality.
To directly answer the question, ALL perjorative words like this are both strong and vulgar. It is ALWAYS advisable to apply suitable non-offensive words (and you can ALWAYS find them), unless you have been offended first and have no recourse - For there continue to be people who only understand and respond to this type of discourse...
Solution 2:
Neither of my grandmothers would have used the word "bitch" except in relation to dog breeding, and perhaps not even there. That said, I know many grandmothers who regularly use the word, and who do not object when the word is used in their presence, that it's safe to say that the status of the word is in transition, and it is considerably less vulgar than it once was. It is frequently used as an alternative for "complain", and takes pretty much the same prepositions and other usage as that word. Further complicating usage is that in some subcultures, "bitchin" is used as an adjective that I understand has favorable characteristics.
That said, I generally refrain from using the word, because over use of the word is obscuring its meaning and impact.