I apologize if someone has already answered similarly since I haven’t taken the time to read all 23 responses that you received so far.
I had a psychologist recommend something that has been very effective at my home (of three boys!). Make your answer be “no” for an appropriate amount of time after the incident. Let me explain. Say that at 10:15am your son does bite his sister. You have put him in a time out, good. (When he is older, after the time out is a good time to review what the appropriate action would be in such a frustrating situation.) Next comes the hard part: say “no” if he wants candy, like “sorry, no candy as a snack: you bit your sister”; and again, “can I go play with my toys outside?” - “no, you bit your sister. You can’t go outside”. “Mom, I want to watch my favorite TV show” - “sorry, you aren’t allowed since you bit your sister.” So, he asks (even non-verbally) for something and you answer with no, because …
Now you don’t make this last all day. You decide when it has been long enough. And, believe me, the next time the biting happens, he may be sneaky enough to get the bite in when you don’t hear a scream; you just find the evidence on the sister’s arm. Make you answer “no” for a longer part of the day.
The psychologist was saying “for the rest of the day”, but after a few times with my youngest missing out on his favorite things for half of the day - I would give in or I would forget about it. 
Be sure, at the end of a time out, to add a minute of discussion of the appropriate response / reaction in such a situation. What you are looking for here is getting to the point where he recognizes that his response could have been to ______ instead of the biting response (I suppose one “appropriate” response could be to move his toy to the other end of the room. Does he have a room or some space that is his space? He will need to be able to “get away” from his sister at times.)
BTW, I’ve heard the “I hate Mommy” and “Mommy, you’re mean” replies. Shrug those off. If you get through on the biting issue, you could start on the saying-bad-things-of-mommy an issue and say “no” for the day! 
And, your child is quite normal to do something over and over again. Sometimes the negative consequence isn’t negative to him or the negative consequence doesn’t seem relevant, to him. What we think is a natural consequence, might not be tied to the negative behavior like we think it is.
Best of luck!