How to Write/Show Psychological Struggle

Generally, you use a character arc to demonstrate a character’s struggle. A personal or psychological struggle would mean that they are both the conflict’s protagonist and antagonist or, in other terms, they are their own worst enemy.
This can take forms like them working off of faulty information or assumptions, making poor decisions or taking self-defeating actions, choosing to be inactive when they should be proactive, or operating on emotions when they should approach a problem logically (and vice versa).

So, like if he doesn’t harm this lady because he has an impulse not to: so what? However, if he should logically harm her (like she poses a threat to him) but his impulse prevents him from doing so, that demonstrates a personal conflict. If he attempts to learn about her in an effort to learn about himself but hanging around her endangers him, that furthers the conflict.

 

Well, there’s an impulse from his human side and an impulse from his vampire side.
The human side, the conflict’s protagonist, is sending the impulse not to harm her through the logical process of “I know and she’s a friend”. The vampire side, the conflict’s antagonist, is sending the impulse to harm her through the animalistic process of “I’m a savage killer so I got to harm her”.

Imagine there’s a little angle and a little demon inside his head piloting him like a giant robot. His thoughts are an argument between the two and his actions are the two of them fighting for control. That would be my advise for writing him. He should probably walking that line between thinking and acting like a rational human or an instinctual monster without going too far into either direction.
Though I guess conventional story structure would dictate that he falls furthest into the monster side around the halfway point of the story while he reaches his most human around the end.

 

You're throwing a little too many plot points at me without me having a good grasp on the story.
I guess to kind of break things down:
- Conventional story structure would imply he dips into his most "antagonistic" (as in, most monsterous) at the middle point. But just because that's the conventional thing, doesn't mean it's necessary or even expected. I'm just saying that while you would ideally have him right on the line between the two extremes, that doesn't mean he can't (at points in the story) dip into one extreme or the other.
- The character has thoughts even if their thoughts are not shown to the reader. You should probably think of the characters decision making process as an argument between his human side and his vampire side. Sometimes the human wins and sometimes the vampire wins. Again he goes back and forth on the two perspectives without fully committing to one or the other (at least through most of the story). These un-shown thoughts would manifest as actions (specifically decisions) which is how the reader understands them.
- Empathy can be a cognitive process as well as an emotional process. That might be worth remembering. Lack of compassion doesn't necessarily mean an absence of empathy.

I guess what I would need to know is that is the vampire side of them a strictly Id-driven emotional illogical and irrational ailment on them while the human side is a strictly Superego-driven logical and moral "true self" that needs to be reestablished as their dominant personality?
Like, is this basically like a werewolf story where the wolfman has to drive off "the beast within them". If so, werewolf stories, traditionally, focus on the character's guilt and remorse for actions taken while in the wolf persona and their drive to lift that curse acts as a metaphor for personal redemption and an absolution of guilt. Often that takes the form of a self-sacrifice for the sake of protecting others from the wolf persona - basically killing the villain by killing themselves - as self-sacrifice is generally considered very heroic but also very difficult.
Depicting your character as unsure in their humanity, insecure or in fear of their monstrous side and having those emotions dictate his choices, perspectives and actions would probably be the proper way to demonstrate the character's internal struggle.

Also: not super-related but you should check-out the movie Martin from 1978. It's a vampire movie, kind of. It's about a guy named Martin who moves in with his older and more religious cousin who believes Martin is a vampire. Martin dismisses the notion, even dressing-up a Dracula to mock his cousin. However, deep down, Martin believes he IS a vampire of sorts as he's compelled to kill women and drink their blood. And his hunt for blood causes a lot of problems, including being beaten-up by one of his chosen victims and being chased by police. He does attempt to give-up on his blood hunt and start a romantic relationship but his psychosis makes him so emotional maladjusted that he can't have a normal human relationship. The whole love story just becomes really sad and pathetic.
But the point is that through the movie, Martin attempts to give into his "curse" and live as a vampire but that causes problems (by putting him in physical danger) while other times he tells himself there are no vampires and lives as a human but that also causes problems (namely causing him despair and misery). Since neither scenario is ideal, he just wavers between the two and that wavering is how his internal conflict is demonstrated.

 

I tend to try if possible to remember an emotion that feels similar to what I'm writing about, or at least is analogous as a starting point. Like knowing a feeling is wrong but still feeling an urge to react according to that feeling. Feeling like I should have more empathy for someone in a situation and exploring the cognitive dissonance involved in that internal conflict. Sometimes acting a bit, getting into the headspace of fear or anger and using that as a starting point. I also feel like having a strong background in practical psychology helps with this kind of thing because it makes it easier to unpack what is going on with a feeling and explain it to someone who might never have felt such emotions. Maybe dig into The Emotion Thesaurus.