Some historians do dabble in psychological interpretations, especially when doing focused biographical work, but it's always fraught and rarely convincing. Why? Firstly, because most historians aren't psychologists. We're literally not trained to do this. The idea that a historian can just read a little psychological literature (or take a few psych classes) and do good psychology is as ridiculous and insulting as the notion that a psychologist could read a history book or two and call themselves a historian. Or that either could read a biology textbook and call themselves a biologist, or a doctor, or anything else that involves expertise. There are a few places where historians are dual-trained, and there are a few works where you have collaborative potential, but by and large this is not common in our field, in my experience.

Secondarily, even if you were the world's best historical psychologist, what really is your evidence-base? Books? Things other said? Diary entries? All of these would be considered gravely inadequate by a modern psychologist to understand a person's internal states. In general modern psychologists think that "long-distance diagnosis" is heavily fraught, and in that they're mainly referring to people who are still alive but they aren't directly working with (hence the APA's discouragement of psychologists from "diagnosing" the current US President). It's hard enough to get inside the head of someone who is sitting in front of you, much less someone who you only know through whatever scraps of evidence history has preserved, much less scraps of evidence they themselves produced with possibly an eye to historical posterity.

Now, these methodological issues have hardly stopped people from occasionally weighing in. Even I have done it here and there, where I felt that a psychological interpretation was useful, and where said interpretation was backed up by an actually-trained psychologist (the one place I've done this is in thinking about Harry Truman in the days after he got word of the atomic bombing attacks, because we do have a lot of confirming evidence from people around him as to his behavior — complaints about anxiety, headaches, dread — and because another, expert psychologist has already suggested that one could read certain common reactions into these symptoms), and even then I am careful to flag very loudly to the reader that I am not a psychologist and this is a very problematic interpretation and the only reason I am making it is because it goes along with my overall argument anyway and as a result it is just a little, somewhat interesting addition to the overall point. Hardly the main driver of it.

I think the broader question to ask is, do historians bring in psychological analysis in a loose way without even meaning to, in the same way they bring in counterfactual analysis whenever they assert something was "important" (which implies that "if things had been different" then there would be significant changes to the direction of history)? Definitely yes, again, especially in biographical work, which is rife with speculation about what motivated someone, how past personal events influenced later ones, and so on. Is this good practice? It's not the most rigorous thing. And it reflects that biographers do feel, after working on a single person so long, that they get "inside their head." But any sufficiently self-reflective biographer knows that this is likely a false feeling and a dangerous one to trust; hence two biographers find a different "inside" even though they've looked at the same evidence.

On the other hand, all historical work is a matter of interpretation, and, it has been argued, a form of autobiography. Historians are not scientists, and though we strive for something that we think has some chance of being "true," we are also aware that we channel the historical interpretations through ourselves (our experiences, knowledge, judgments, etc.), and that results in a certain degree of unavoidable subjectivity. Some embrace that subjectivity, some try to deny it, but it's always there. It doesn't mean we don't produce useful work. It doesn't mean that "anything goes." But it does mean that the work always bears, at some level, the stamp of us, the historian, with all of the positives and negatives that brings with it. And if that includes our understanding of human behavior, and (informed!) assumptions we make about how a given mindset might have come about, well, so be it.

Should we appeal to psychological factors? Probably not — at least, not without a lot of good reason to do so, and always with humility. Do we? Sometimes, sure.

Generally speaking, no, though that doesn't stop people from trying. To understand someone on that level you basically need their diary, or at least a large number of their private letters published without editorial oversight. Even then, you have to hope that they were nice enough to tell you their most intimate thoughts in a level of detail that is actually useful. You've given Caesar and Alexander as examples, so I assume your interest in is antiquity, and whilst the psychology of those two figures somewhat eludes us, it is possible to delve into psychological motivations for a couple of ancient figures. Cicero is one, as we have a large number of his private letters and he was pretty open about his emotions. The other is Marcus Aurelius, as the book of Stoic philosophy called Meditations is literally a journal of his thoughts. Other than those two figures though, nobody comes to mind where you can make a confident assessment of their psychological motivations. Cicero is convenient enough to have written a lot of blunt literature on his state of mind. Marcus Aurelius is more difficult, because his thoughts are heavily wrapped up in philosophy, but he is similarly honest and blunt.

Part 1 - Cicero

Cicero spent significant periods of his life depressed and at times was suicidal. It is extremely rare that any historian can confidently make a statement about the mental illnesses of a historical figure, but Cicero is an exception. In the 3 years or so before the death of his daughter Tullia, his mental health was in decline anyway (Cicero seems to have struggled with depression for much of his adult life), but he completely broke in 48 BC and it is in letters from 48-45 BC that we get the most insight into Cicero's struggles:

Extract from a letter to his brother Quintus (1.3):

The fact is, the much-praised consulship of mine has deprived me of you, of children, country, fortune; from you I should hope it will have taken nothing but myself... [Cicero apologises extensively for not writing in the weeks following his daughter's death]... as to the slaves coming to you without a letter, the real reason (for you see that it was not anger) was a deadness of my faculties, and a seemingly endless deluge of tears and sorrows. How many tears do you suppose these very words have cost me? As many as I know they will cost you to read them! Can I ever refrain from thinking of you or ever think of you without tears? For when I miss you, is it only a brother that I miss? Rather it is a brother of almost my own age in the charm of his companionship, a son in his consideration for my wishes, a father in the wisdom of his advice! What pleasure did I ever have without you, or you without me? And what must my case be when at the same time I miss a daughter: How affectionate! how modest! how clever! The express image of my face, of my speech, of my very soul! Or again a son, the prettiest boy, the very joy of my heart? Cruel inhuman monster that I am, I dismissed him from my arms better schooled in the world than I could have wished: for the poor child began to understand what was going on.

Extract from a letter to his brother Quintus (1.4):

I beg you, my dear brother, if you and all my family have been ruined by my single misfortune, not to attribute it to dishonesty and bad conduct on my part, rather than to shortsightedness and the wretched state I was in... The tears of my friends prevented me seeking refuge in death, which would certainly have been the best thing for my honour, the best escape from unbearable sorrows... I shall retain my life as long as I shall think that it is in your interest for me to do so, or that it ought to be preserved with a view to future hope.

This level of grief was not normal for the day. From the death of his daughter in 48 BC to late 45 BC, a lot of Cicero's letters are overwhelmingly negative, self-loathing, self-hating, and largely devoid of happiness or the sense of humour typical of his other letters. He had good days of course, but most of the letters show a man seriously struggling with life. From then on Cicero suffered bouts of depression, usually related to the tumults of his post-Caesar political career.

To quote from the article "A grief observed: Cicero on remembering Tullia" by Han Bultussen:

Due to his personal circumstances he found himself in total isolation suffering on so many fronts that he ended up in a depressed state, resembling pathological grief. Thus we can explain the obsessive nature of his responses.

And from "Interrupted by fits of weeping: Cicero's Major Depressive Disorder and the death of Tullia", published in the journal History of Psychiatry by Kathleen Evans:

the wealth of material revealed in the letters about Cicero's physical and emotional state enable a diagnosis of Major Depressive Disorder to be made with some certainty according to the DSM-IV-TR. Cicero appears to have experienced increasingly severe bouts of suicidal depression that seriously impaired his relationships with his friends, family and political colleagues, and possibly shortened his life.

What Evans argues (and I should point out that she is principally a psychologist and not a historian, when usually with this kind of thing it is the other way around) is that Cicero's mental illnesses meant that he struggled to maintain political relationships, as he was prone to pathological episodes (such as his obsession with building a shrine to his daughter) which alienated potential allies and made the political elite frustrated with his behaviour. The implication of this for a historian is that it explains some of Cicero's actions following the death of Caesar. Cicero was the leading man of Rome and many senators looked to him for leadership of the Roman Republic. It can be argued that his poor mental health at that point in time meant that he failed to capitalise on the opportunity, which gave Marc Antony and his allies room to act against him.

The other thing about Cicero is that he was insecure. His family had almost no political pedigree and he was termed a novus homo - a "New Man". In other words, he had something to prove. He wanted to prove that he was just as worthy of being a senator as people like Pompey, Caesar, or Cato the Younger. He often tried to hide this, but it shows itself quite clearly in some letters, such as this one to the historian Lucceius.

I have often tried to say to you personally what I am about to write, but was prevented by a kind of almost clownish bashfulness. Now that I am not in your presence I shall speak out more boldly: a letter does not blush. I am inflamed with an inconceivably ardent desire, and one, as I think, of which I have no reason to be ashamed, that in a history written by you my name should be conspicuous and frequently mentioned with praise.

Cicero does not wish to admit it and pretends like it's totally normal, but he had clearly been worrying a lot about what Lucceius had to say about Cicero's political career, and in the letters I quoted to his brother, we can see a serious insecurity about whether he had actually done more harm than good as consul. Whilst Cicero also had moments of great happiness and confidence (see his triumphant return from exile, for example), it is also quite easy to see how his insecurity and mental illness may have contributed to his political actions and his writing. He arguably overreacted to the Cataline Conspiracy, for example, in which Cicero illegally executed conspirators without trial. It can be argued (and often is) that he did this because he wanted to be seen as acting decisively, like a true leader, in part because he was insecure about his leadership abilities and what people thought about him.