Master: "You must have a death wish just to talk to me, Master. You know how dangerous I am, right? I'm meant to kill your enemies, not socialize. If you're not careful, you might end up dead in an alley with your guts strewn everywhere… Eh? You still want to talk? My God, you must be madder than me."
Avenger's relationship with her Master is… complicated. She's tired of her Master's attempts to connect with an irredeemable murderer like her and has to threaten them multiple times to stay away.
Although, she is fascinated by her Master's persistence. Perhaps she's impressed by their bravery, or stupidity as she calls it. Why would they summon her in the first place anyway? Avenger is far from a Heroic Spirit, and yet… they still summoned her. At least for now, she'll play along with her Master and be as nice as she wants to.
Remember though, this is the same woman that brutally murdered 5 women and sent London into a panic. Tolerable she may be, her treatment of her Master is more akin to a predator playing with their food to satisfy their curiosity and boredom. It is required that the Master does not let their guard around her.
Holy Grail: “Every day I think about the young innocent souls disregarded by this cruel world. I can't stand to see their despair anymore. I'm going to make sure every child that ever lived will be loved, then my purpose will be complete.”
Avenger wishes for the salvation of all abandoned children in the world. Some people may think she would want to wish for her children or something similar, but she fears that she will lose them again. Instead, she wishes to help the children who have nothing left. She knows she can't save everyone by herself, so relies on the miracle the Holy Grail can grant. She will be determined to win the Holy Grail and will work with a Master as long as they follow her sense of values.
Assassin - Jack the Ripper: "You… I- I'm sorry… You're everything I wished to prevent… I don't want to see you suffer anymore…"
Out of all the child Servants, this one, in particular, hits Avenger the hardest. Their circumstances are so different, yet so similar. In a sense, Jack is the embodiment of Avenger's guilt and fear, made to taunt her about her failures. Instead of anger, Avenger feels remorse and pity towards Jack, blaming herself and the world for making Jack suffer.
With Jack, she sees Avenger as the ideal mother regardless of her confusion why Avenger seems so apologetic towards her. She's almost convinced that Avenger is her actual mother supported by the similarities of their appearances. So it is understandably shocking for Avenger when Jack willingly accepts her as "mother." Jack may be the only person that makes Avenger reconsider her "path of vengeance."
Even with the irony of sharing the same identity, both of them are tragic souls made for each other. They may be the only two that work efficiently together; both as a combat duo and a "mother-daughter" pair.
Berserker - Jack the Ripper: “How many of us are there? First Smaller Me, now this bloke… For God’s sake, it seems like everyone can become Jack now. What’s next? A magical gi– oh.”
Avenger does have a sense of respect for the Berserker - Jack the Ripper. She is impressed by the range and potency of this Servant's power. The problem with Berserker she is annoyed and confused by is how many versions of Jack exist. Berserker's shapeshifting and cloning Noble Phantasm doesn't help at all, leaving Avenger to wonder which is the real Jack.
While Jack knows she shouldn't be surprised at the variations of the same person, she thinks it can get annoying that a group with the same name is running around, confusing people when referring to Jack.
Caster - Bluebeard: "▂▂▃▃▄▄▅▅!"
Words aren't even needed to describe Avenger's hatred towards Bluebeard. Chances are she will actively try to kill him and anyone like him as brutally as possible.
Archer - Atalanta: "Your wish… it's just like mine. Even with all our differences, our dreams are the same. Heh, at least I finally met someone in the world that cares."
Despite Avenger being a serial killer and Atalanta a hero, both at least acknowledge and understand each other's wishes. They find kinship in their desire to protect and save the children in the world, even if they butt heads sometimes due to conflicting personalities. Nevertheless, they have one of the strangest, but understandable friendships to date.
However, if they are forced to kill each other, they will do so as they have confidence in each other to fulfill the one wish that drives them.
Caster - Irisviel von Einzbern: "So, this is what a happy family looks like. Even through hardship, you faced it all with the people you loved and pulled through. I don't understand how you all could still stay happy after everything that happened to you. I… really envy you."
With Irisviel von Einzbern and the Emiya Family, Avenger considers them what a model family should be. Even with all their faults, their bonds remain stronger than ever.
At least, that's what Avenger thinks.
Avenger puts them up on a pedestal that she disregards any other doubts or flaws they may have. It may be because she is this desperate for something good in the world to exist. She believes that everyone, child or parent, should have a family just like the Emiyas.
In her eyes, Avenger sees what she might've had if fate had been kinder to her. This kind of admiration is a double-edged sword, as it turns to jealousy. She imagines herself in Irisviel's place many times, dreaming of having the family she would never have. But in the end, it is a selfish dream born of jealousy and hope.