For eighteen years, Voldemort lived in a Muggle orphanage where, for almost twelve of those years, he did not know who or what he was:

He was a funny baby too. He hardly ever cried, you know... (Mrs. Cole)

During infancy, crying is natural and important for development as it is a biological reflex that allows a baby to release distress from internal displeasure (hunger, pain, discomfort), but is also vital for survival since it is the only means a baby has to ask for help from a caretaker. The most important experience that forms is this: unattended need -> crying -> help -> needs attended. It creates a cycle of trust and bonding that later serves as a foundation for emotional attachments with adults and other children.

Voldemort being a baby who "hardly ever cried" can be a concerning sign that from an early age he might never have learned how to ask for help to fulfill his needs and desires, but also to form attachments, which might have also influenced him to resort to scaring the other children and stealing from them instead of borrowing or playing together.

... when he got a little older, he was... odd... He scares the other children... (Mrs. Cole)

For a child, stealing is quite a natural behavior. It can be a way to test the limits of the caretaker's affection and patience; an attempt to repair a feeling of deprivation and loss, especially if the stolen object acts as a symbolic substitute for love; or even a more unconscious way to manage guilt.

.... New Year’s Eve and bitter cold, snowing, you know. Nasty night... We took her in, and she had the baby within the hour. And she was dead in another hour. (Mrs. Cole)

What Mrs. Cole told Dumbledore is likely what Voldemort heard as a child, since children who grow up in orphanages do ask others and themselves what could have happened that led them there. They understand that something down the road went differently from other children who are raised by their families. Voldemort knew he was born on the same "nasty night" - cold and bitter - that his mother died. And since births and deaths were not meant to meet, I can only wonder how he felt about this.

It's not unusual that children who lost their mothers to childbirth often internalize misplaced guilt and believe they have caused their mother's deaths by being born. This requires caretakers to be clear and open - and much more gentle than Mrs. Cole seems to have been - to explain that maternal death is never a child's fault. I've found no canon support to defend this as an affirmation, but I wouldn't be surprised if Voldemort struggled with feelings of guilt or fault over Merope's death during his early childhood, which would make his grieving process more difficult. The point is that, as a child, Voldemort seems to have understood that death was painful:

... Billy Stubbs’s rabbit... it didn’t hang itself from the rafters, did it?... All I know is he and Billy had argued the day before... (Mrs. Cole)

Now, I won't defend that animal cruelty is natural child behavior, however children don't simply kill or torture animals for "fun", those actions usually also have other meanings. Analyzing this would require confirmation of his age, but, since it's implied he used magic, I'd guess this likely happened between the ages 5-11. This might have implications such as trauma reenactment, early displays of sadism, or even restrained empathy, but there is no way of knowing for sure without his canon age.

The important point here is that Voldemort had enough understanding of death to, when annoyed because of an argument, kill something Billy loved because he knew death could hurt more than a physical fight, an insult, or even simply stealing from him. Voldemort chose to use death as a method of stealing.

I don’t think many people will be sorry to see the back of him... [Him returning every summer would be] better than a whack on the nose with a rusty poker (Mrs. Cole)

Regardless of their motives, Voldemort was a child whose presence was perceived as unwanted and unwelcome. We don't know for sure if this was ever something that truly bothered him, but overall this isn't an easy feeling for a child especially because it would add to his abandonment trauma. Another aspect of this is that this is likely a variation of what Voldemort heard when he asked about his father and other relatives:

... no Tom nor Marvolo nor any kind of Riddle ever came looking for him, nor any family at all, so he stayed in the orphanage and he’s been here ever since [his mother died]. (Mrs. Cole)

Children tend to resort to fantasies, often innocent and hopeful, to fill in the gaps of what they don't know. What sort of places Voldemort's imagination led him as a result of no one ever seeking him out it's up for us to wonder, but this is what an eleven year old boy concluded, after discovering being a wizard, as the question "burst" out of him:

Was my father a wizard? He was called Tom Riddle too, they’ve told me... My mother can’t have been magic, or she wouldn’t have died... It must’ve been him...

Right here is an orphan child asking about his father in "spite of himself", as if it had been building inside him his whole childhood, and saying "more to himself than Dumbledore" the conclusion he arrived to finally put meaning on his mother's death:

Magic should prevent death

Magic has become associated with his childhood trauma, which was centered around mortality and death and therefore became connected with his grief. Immortality now becomes much more about this little child who did not understand that death, tragic and harsh as it can be, is something natural and expected instead of feared and conquered. And what he became is the result of a grief that I believe is beyond the capacity of an eleven year old who never learned to ask for help. Consider this reaction:

What are you here for? Did she get you in to have a look at me?... She wants me looked at, doesn’t she? Tell the truth [...] You can’t kid me! The asylum, that’s where you’re from, isn’t it? 'Professor,’ yes, of course — well, I’m not going, see? That old cat’s the one who should be in the asylum. I never did anything to little Amy Benson or Dennis Bishop, and you can ask them, they’ll tell you! [...] [When Hogwarts is mentioned as a special abilities school] I’m not mad!

For a child to have such an understanding of what an asylum and "mad" even is, again, concerning. In the 1940s, mental health was far from what we have nowadays. I'm talking about massive and overcrowded institutions that used what they called treatments back then, which can now be compared to methods of torture. And yes, it did include children his age.

I wouldn't be surprised if the "asylum" was a threatening tone the orphanage workers used to make him behave, or even an effort they made trying to get rid of him. Voldemort also perceiving "professor" and "doctor" as roles to be feared also speaks about his avoidance of help as both figures are associated with power dynamic, but also as professions we search when in need of guidance or healing.

I don’t need you... I’m used to doing things for myself, I go round London on my own all the time...

I fear repeating myself, but, again, concerning. What I'm trying to set here is the extent of Voldemort's denial of help or needing others. Despite believing that the Muggle war did play a role in his future pursuit of power and added to potential feelings of being helpless and unprotected, I'll avoid bringing this subject to not exhaust this reading. So far, Voldemort's childhood has been building the wizard who perceives needing help and trusting others as the ultimate weakness.

When Voldemort arrived at Hogwarts, we know he searched for information about his father, believing he would be a wizard. Then, soon before or during his 5th year he discovered not only that his mother was a witch, but that he was a descendant of Slytherin himself, so Merope had the power to survive, but somehow did not. I'd set this as the probable moment where his childhood realization from earlier became:

Power can prevent death

I'm not saying he wasn't interested in power before, but discovering his mother died as a witch probably was a painful truth to realize because once he learned that she had magic and still died anyway, the childhood problem that teenage Voldemort had to solve psychologically became associated with magical power.

Voldemort grew up knowing that his mother died an hour after he was born and all she could give was his name, the same from the Muggle man who had abandoned both before his birth and because of magic:

My mother died just after I was born, sir. They told me at the orphanage she lived just long enough to name me — Tom after my father, Marvolo after my grandfather.... ....You think I was going to use my filthy Muggle father’s name forever? I, in whose veins runs the blood of Salazar Slytherin himself, through my mother’s side? I, keep the name of a foul, common Muggle, who abandoned me even before I was born, just because he found out his wife was a witch?...

A name Voldemort abandoned when he became a powerful wizard, just as his Muggle father abandoned him before birth when he discovered Merope was a witch. There is an interesting connection with magic and abandonment in Voldemort's life:

... My mother, a witch who lived here in this village, fell in love with him. But he abandoned her when she told him what she was... He didn’t like magic, my father... He left her and returned to his Muggle parents before I was even born, Potter, and she died giving birth to me, leaving me to be raised in a Muggle orphanage

For a psychoanalytic reading, whether or not Voldemort was telling the truth, it does not matter here. This is what Voldemort himself believed and, even if he lied, this is what he chose to lie with and therefore is his truth, which might be even more revealing than what is canonically true.

If Voldemort's childhood conclusion was that magic should have prevented his mother's death, then discovering she was a witch who died transformed the problem he'd attempt to solve in his teenage and adult years. Rather than accepting death as natural to being human, he crafted a whole new identity around avoiding this realization.

In a way, power became his means and immortality his goal as both were psychologically operating as symbolic substitutes for mourning and accepting Merope's death and a defense against his early childhood traumas: losing his mother and being abandoned by his father, both events tangled with magic from his perspective. For adult Voldemort, conquering death might mean the loss that had shaped his whole childhood would never have to be experienced again.

Why, then, was his goal to make himself immortal?

Psychoanalysis has two core concepts that might help explain.

During infancy children experience relationships through primary narcissism which is fundamental in early development as babies invest their energy (libido, if you will) in themselves. For an infant, the world revolves around them, which is also why those early unattended need -> crying -> help -> needs attended experiences are so important. This is completely natural as a developmental stage where the distinction between the self and the external world is still forming. Later, as the child develops they start to realize that, well, the world does not revolve around them, and then secondary narcissism starts to develop and the child now invests both in themselves and in others. This creates a balance between investment in the both the ego (ego-libido) and in others (object-libido).

Then, the concepts of ideal ego and ego ideal (both connected with the self) refer, respectively, to an idealized image of the perfect self and the standpoint from which a person measures and judges themselves. Becoming Lord Voldemort - a powerful, immortal, feared and self-reliant wizard - can be interpreted as his way to bring the symbolic death of the abandoned and vulnerable child he once was.

More than a change of name, Voldemort constructed a new identity organized around an idealized image of himself that leaves no room for dependence, vulnerability or mortality. In a way, even the murder of his father can be understood as an attempt to symbolically kill 'Tom Riddle' as his former self. Immortality is not simply an ambition for Voldemort, it's a psychological necessity because an ideal self that remained mortal would still be vulnerable to the very thing he grew up feeling helpless against: death.

Because Voldemort's early trauma revolves around abandonment (by death or rejection), his capacity to emotionally invest in others may have been compromised. In a way, instead of investing in external objects, Voldemort usually invests and reinvests what he gets from himself and others back to his ego. Since emotional investment in others is unconsciously associated with abandonment and death, even his ultimate goal of being immortal to prevent loss became, in a way, another investment in himself.