To understand Godot, both sides of his character have to be looked at: who he is as Diego Armando, and who he becomes as Godot. While Diego is always the heart of who he is, Godot serves as a mask and a crutch, allowing him to simultaneously acheive his goals and escape a reality he doesn't want to face.
First, we have Diego. Confident to the point of smugness, he can be a touch on the rude side to others, typically referring to them by nicknames based on their apparent characteristics. He tends to be demanding of people, and generally respects those who have guts, even if they're shown through foolish acts. But beneath the bluster, Diego can also be a kind person, in his own way. He generally knows how to put on the charm (when he wants), and he also quite willingly acts as something of a mentor at times, be it through subtle hints or outright pushing. Notably, Diego usually says whatever needs to be said at that moment--meaning he's quite good at reading people and situations, though his bluntness may not make it seem that way. On top of this, Diego is a man who loves his metaphors, which only makes it all the harder to understand him sometimes, but that also usually relate to the topic at hand.
One of Diego's major problems is that he has a tendency to fixate on certain things--to put it lightly. A good example is coffee. Diego loves coffee with the power of a thousand burning suns. He's made at least 107 unique blends, and he knows that there are 253 distinct types of bitterness to coffee. He also seems to have based most of his life philosophy on coffee--its color, its taste, its scent, so on. And he practically drinks it by the gallon.
The most prominent trait about Diego, however, is his steadfast adherence to morality. More than once he claims a statement to be one of his rules, and he not only abides by it himself, but expects others to do the same. To this end, he again won’t mince words, and he also won’t back down when he wants to see justice done. And ultimately, it causes Diego to create Godot.
The decisions Diego makes after waking up are pivotal to his character. As he said later in the game, when he woke up, his best days were gone, and there was no one waiting for him. He could neither devote himself to love or revenge, thus finding himself in a desperate situation. He can't deal with the pain of it, so he sought something else to focus on, and someone else to blame for Mia's death. That he holds himself responsible despite his being in a coma says a great deal about his character. Diego is a protective type, clearly, and he won’t easily let go of anything he holds himself responsible for.
Unable to cope, Diego finds two new things to focus on. The first is protecting Maya Fey, Mia's sister. This isn't out of caring for Maya specifically, but because Maya is the person Mia loved most. By guarding her, Diego believes he can somehow make up for letting Mia die. The second thing is a jealousy-routed hatred of Phoenix Wright. Diego can't cope with the life he's found himself in. Yet Phoenix, Mia's former client and junior partner, has prospered after Mia's death. Diego fixates on Phoenix, transferring the blame he feels towards himself onto the other man-he was there, he could have protected her!
Yet these fixations go against Diego's core of rationality; he ultimately knows the truth, even if he can't accept it. So in order to be able to move on, he creates a new identity: Godot. Again, while this identity is not totally separate from Diego, it is distinct. Ultimately Diego takes responsibility for the actions he takes as Godot. But as this new persona, he is willing to go against his basic principles in order to achieve his goals. Godot becomes a prosecutor, the polar opposite of a defense attorney in the Ace Attorney universe. While the defense trusts, the prosecution doubts. The defense protects, and the prosecution attacks. And as a prosecutor, Godot can go against Phoenix in the arena they both know best: the courtroom. He wants to see what kind of man Phoenix is through his actions-and unconsciously, Diego is studying the differences between them.
As Godot, he also breaks the law, which he would have never considered before, by committing murder and setting up someone else for the crime (albeit with permission). For a man who lives by a set of rules, and who prides himself on those rules to such a degree that he also expects others in similar position to abide by them, such violations of moral principle are all but unthinkable. Yet as Godot, he is able to do it, using his talents and intelligence to become the kind of person Phoenix would have to work hard to defeat. And that is the purpose of the identity; to live for the only true reason Diego thinks he has left. At the same time, Godot also is Diego's way of punishing himself. As said, Diego doesn't deny responsibility for his actions. When he is accused of murder, he repeatedly asks for proof-but he never says he didn't do it. Those unthinkable measures had to have been a heavy burden on Diego's soul, and they ultimately lead to his arrest due to Phoenix's efforts. If Godot didn't exist, the murder wouldn't have happened, and Diego would have never been appropriately punished for the crime he felt most guilty of: causing Mia's death. Godot ultimately is Diego's method of finding salvation when he thought it impossible, though he denies this truth to himself until the very end.