Roland Auguet's biography of Caligula deconstructs the common myths about him. The actual data available points to how he was very popular among the actual Roman people and his policies were reasonable and made with real intent to improve the well-being of the citizens of Rome. However as you said he absolutely hated the Senate, and he kinda had a good reason to do that since thanks to the Senate his beloved mother was made an outcast, viciously beaten, had her eye gouged out, and finally was starved to death. This would probably make most people upset, so it's understandable Caligula took every chance he had to humiliate members of the Senate.
There's some degree of discussion regarding how much of an actual lolcow Caligula was. If I remember correctly the ones writing Roman history in these days were affiliated to the Senate, and Caligula absolutely HATED the Senate and defied them whenever he could. So it's not unlikely that they exaggerated the bad things he did to make him look worse than he actually was.
(There's also how he seemingly started more or less okay but, but few months after his reign began he got severely sick and THEN went loopy. Hm. )
tl;dr Caligula did nothing wrong