I guess "the hate" comes from the extreme arrogance that Hamilton puts on display, and his team boss does a great job on being second best arrogant in the team. Remember "the hunger" interview when he gave the reasons why he would win against Nico Rosberg? This hunger came from someone who dates Pussycat members, swims in money and records his own music in a state of the art home studio..? The Mercedes team and Hamilton seems so entitled to winning that it is natural to want to see them fail. The only seem likeable and approachable when the PR team has done their job..
Now that being said, i think that Hamilton has issues, not surprising since he hardly had any chance to live anything close to a normal life. And i am pretty certain that somewhere in there behind his defensive arrogance is a guy that can be a cool guy... The difference to Rosberg is spectacular, Rosberg does commentary with the german tv station RTL, he is relaxed and a cool dude, totally in balance with himself, today he met Hamilton, but Hamilton didn't "see him". He ignored him and looked the other way and try to avoid him
You say arrogance, I say it is all a matter of personal perception and/or what people don't like about him or his personal life choices. For me, I could care less about the personal life of the drivers because what I'm watching is Formula 1 Racing, so whatever happens on the track is the only thing I care about. Am I entertained, was the racing good, was it another processional race, did the drivers put on a good show, is F1 going downhill or is it still exciting?????? That's all that matters to me.
Also, as far as I can recall, every top drivers on that grid has displayed that "extreme arrogance" you mentioned in some way or another. When Vettel was winning left and right, he's had his moments. Same can be said for Kimi and Alonso at diverse point of their career. Heck, you mentioned Rosberg, I'd even put him in there too (especially when things were getting very heated between him and Lewis), he's not racing in F1 anymore so it's very understandable for him to be relaxed and cool. I don't recall seeing him like that when he used to race against Lewis, maybe in the instances where he won. People seems to be OK with it coming from those guys (nah it's just an emotional outburst) but they can't stomach it coming from Lewis.
In any case, everyone is entitled to their opinions and I'm in no position to tell people who they can and can't hate. All I'm saying is that, FANS OF THE SPORT, regardless of who you support, like or hate should give the drivers the credits where credits are due instead of downplaying (it's the car, it's a top team with tons of money, it's this or that) their achievements because they "hate" them. I'm not a fan of Lewis but for once in the last 3 years I actually want him to win this season just because of that.