This is hotly debated, and there are two issues wrapped into your question. The first is: did Caligula really do the crazy things he is said to have done. The second is: was Caligula suffering from mental illness. Historians thus fall into four camps: neither, one but not the other, or both.

The reason why this is debated is because there are clear reasons for Roman historians to give biased portraits of Caligula. First is because making Caligula look terrifying makes Claudius look good and absolves his killers. Second is that making all the Julio-Claudians (Augustus and his successors) look bad makes Vespasian look like a good change (he became emperor following a civil war after the death of Nero). The sources we have for Caligula were all written after Vespasian. This means historians have to take a cautious approach to the sources and ask questions about every bit of detail: how would the author know this bit of information? How might this event be "spun" for political reasons? Is the author drawing pointed comparisons more for emphasis than for fact?

I believe that Caligula did many of the crazy things he did but that he was not actually mentally deranged. A lot of the stories we hear about him smell of spin and artistry. Whipping the English Channel is a reference to Xerxes the Persian king of kings who invaded Greece. Ditto the stories of him comparing himself to gods and taking his sisters for wives. Those are Egyptian practices. I read these more as the source's attempt to paint Caligula as an Eastern despot as opposed to a proper emperor (Augustus or Vespasian or Trajan).

Then there are the stories of Caligula's incredible feasts and his goading of political "elite". I like the interpretation that these are ways for Caligula to flaunt his power over the senatorial class. Because Caligula was so young when he came to power he did not have time to legitimize his role in government by holding offices and commands like Tiberius. Add to that that this is only the second time someone had succeeded to the principate and you can see that Caligula needed some way to promote his claim to primacy in Rome. He did this in the most brash way imaginable: rubbing his wealth and political power in people's faces. He made a horse a senator to show the actual senators how little their office was actually worth. He spent so wildly on feasts so that even the wealthiest people in Rome felt like paupers. He cajoled and shamed Romans in order to prove his place at the top.

Here's the issue: this argument has internal logic but it's essentially unproveable and it goes a bit against the primary sources (who claim that Caligula was mentally unsound). However if we are critical of the primary sources as we ought to be then I think there is room to look at Caligula as a rational actor in a tough position.

Let me leave you with a bit of a thought experiment. Imagine if Donald Trump's political supporters were the only non-government source for Obama's presidency. A historian would only have their word and the official records like signed laws and executive orders from which to build their biography. We would probably end up with a story about a Muslim megalomaniac who became the first foreign-born president of the US. I don't mean to draw any comparisons between Obama, Trump, and the emperors, but that situation is exactly the issue we have with our sources for Roman emperors.