Personality
Gundham has an eccentric and somewhat dark personality, his delusions of self-grandeur likely stemming from an inferiority complex and pseudo syndrome known as chuunibyou. He often claims that he has been blessed with the evil powers of the underworld, refers to everyone else as "pathetic human beings" and claims that even the gods themselves fear his power. He frequently rambles on about being the "Supreme Overlord of Ice" and an offspring of an angel and a demon with an interest in things like the dark arts, alchemy, and astral levels. He is often lost in his own little world, preferring to be alone and usually paying more attention to himself or his hamsters than the things that are going on around him, such as in Chapter 1 when he failed to investigate after the murder and putting his lost Hell Hound Earring at a higher priority. As a result, he can come across as rather uncaring and self-centered.
Despite first impressions, Gundham is generally rather comical in what he says to Danganronpa spectators (not so much for characters actually talking to him) due to his behavior resembling an overly dramatic cartoon supervillain. Humorously, he sometimes even compliments other characters if they say a "nice line". He can be somewhat contradicting or counterproductive—in his introduction, he tells Hajime Hinata to step forward and introduce himself, for Gundham will destroy him once he does. It is heavily implied that though he puts on the face of a loner who wouldn't dream of hanging out with pathetic mortals, Gundham actually may be very lonely as he once boasted that he was searching for solitude before wandering away slowly and silently. Furthermore, despite his dramatic and seemingly confident personality, he is actually quite shy when complimented or if he has a more intimate moment with someone, as he tends to blush a lot and quickly hides his face with his scarf. It also appears that he has very genuine and serious issues with social events and physical contact, as he has been very strict about not wanting to touch nor be touched by other people since he was a child, going as far to avoid certain activities altogether as a result. However, in his usual manner, he claims that he avoids physical contact because fatal poison runs in his veins and he can only be touched by very specific people with matching astral levels. He eventually gets better with both social and physical contact.
Despite his issues with people, Gundham genuinely cares about animals deeply (though he often refers to them as his evil minions due to his overlord persona), and understands them much better than he understands people. He claims that anything that can talk will eventually betray you and he highly respects the sincere nature of animals. He demands that animals must be treated respectfully and scolds people who fail to do that, but simultaneously wishes to educate and give advice to those who don't know how to treat certain animals correctly. In his Free Time Events, he tells Hajime that he runs an informational website about breeding which he maintains carefully, and he states that he pities livestock for being raised for the sole purpose of human consumption. This could mean that Gundham is possibly a vegetarian or vegan, though it is not confirmed, and his words may imply that he is only against eating animals when there is lots of human involvement he considers unnatural. He takes a great offense whenever humans intervene with animals' natural behavior and control their lives too much. In Island Mode, it's revealed that he even straight-up hates anthropomorphic cartoon animal characters, likely considering their human-like behavior extremely insulting towards real animals.
In general, Gundham has great respect for nature and its laws. He understands that nature can sometimes be harsh and cruel due to constant fighting for survival, and thus he does not have any issues with death, but he strongly dislikes the humans' methods that are often truly malicious and very disconnected from the natural order. Notably, he strongly dislikes the idea of giving up without a fight, as he believes that all creatures by nature fight to their last breath in order to live, and that giving up your life willingly would be a blasphemy towards life itself. When the students are starving and trapped in the Funhouse, Gundham and Nekomaru agree to duel to determine which one would be a sacrifice to get everyone out. Gundham ends up killing Nekomaru and when he's exposed as the murderer, he doesn't put up much of defense nor seem to be that bothered by his actions, as he believes that fighting for survival and sometimes even killing another is necessary and natural. He later points out that Nekomaru could have run or called for help, but instead chose to stand his ground against him. However, after Nagito suggests that he took Nekomaru from behind, Gundham is absolutely furious at the claim that he'd do something like that.
While he claims he is only motivated by his own will to survive, it is implied that Gundham may have been conflicted about sacrificing his friends and was partially motivated by his will to save them, though he, of course, denied this fiercely while relying on his overlord persona. Earlier, he even gave a rousing speech about the importance of hope and not giving up on life even in the face of despair, and thus nobody is really convinced by his declared motive. Nevertheless, it seems that he was also ready to let the others die according to his ideology (though it would've likely caused him guilt) if that was how the events turned out. While he showed some signs of panic earlier during the trial, in the end, he accepts his fate with dignity and stands up confidently against his execution, with his only worry being the well-being of his hamsters. It is implied that his "evil" persona was overcompensation for his underlying, gentle spirit by creating an entirely new, exciting personality. Despite talking about going to Hell, his execution scene shows animal angels carrying him to Heaven, once again indicating that his true nature is good.
Interestingly, Gundham has moments when he shows a deeper understanding of certain situations, such as being the only one who openly acknowledges that Nagito's way of thinking is grey morality (despite strongly disliking the other boy), and he states that murder is murder regardless of intentions.
While there isn't much information on his brainwashed self as the Ultimate Despair, it appears he used to lead intimidating armies of wild animals, likely possessing a danger to humans. He also kept his eyes shut and wore a straitjacket, which might symbolize the spiritual and emotional state he possessed at the moment. During his high school years, he only referred to himself as "Tanaka the Forbidden One", but he proudly proclaims his true name and identity after he is awakened from the coma with the brainwashing's effects removed.