I also have that doubt before, but after reading his post I believed it is really him.
The sites are post on Chester's official website. I know you can argue that it's a fan site before and someone manage for him.
He posted some private pictures that I don't think you can find on the internet.
The conversation between his wife, Chris Cornell, and Chris Cornell's wife make their twitter look more legit, since Chris and his wife's Twitter are offical.
I 90% believe that it's really him. 10% wait for the official announcement on Linkin Park.com.
I think he is addicted to twitter.
Crazy man................