Alice and Ace have a very unique relationship in that they both commiserate together. Ace loves Alice being lost, so he doesn't try to help her 'lostness', and Alice has too much guilt to change without other people influencing her. In the end, they both don't really understand why they like each other how why they are together, but they do still like each other. In a more literal sense, Ace and Alice's relationship (in Joker especially) is more so like a 'friends-with-benefits' situation to start with and progresses more as the plot unfolds.
Ace is indifferent toward Blood and Blood appears to be outwardly the same toward him, though Blood acknowledges that Ace is a formidable opponent who he would rather not fight.
Ace and Elliot have a strained relationship. While Ace does not appear to hold any particular dislike toward Elliot, Elliot dislikes Ace due to him always coming onto the Hatter territory claiming it is a shortcut to wherever he wants to go.
Tweedle Dee & Dum
For the most part, Ace doesn't mind Boris; they don't interact much overall. If Boris is perceived as a threat or "in the way," Ace will attack without hesitation, but their interactions are generally more nonchalant.
While Ace doesn't really like very many people, he truly hates Pierce, because the Dormouse's job makes Ace's, and by extention Julius', job more difficult. As part of Pierce's job is to "hide the clocks" that Ace gathers, he works in exact opposition to Ace, who gathers the clocks. Ace tries to kill Pierce whenever they meet.
Ace does not respect Vivaldi. He calls her an old woman and insults her to her face while laughing. Because she is his superior, he can't do anything more than that, but like Peter, he would betray Vivaldi for Alice.
Ace claims to look up to Peter, calling him "Mr. Peter" ("Peter-san"), while Peter only looks down on Ace. Peter will try to shoot Ace at the slightest provocation, which never has any effect. Both admit that the other is strong, Peter going so far as to say that Ace is fearsomely strong. Ace says that when Peter is acting sweet and adoring toward Alice, it's unsettling and even disturbing; the side of Peter that Ace likes is his heartless side.
Ace calls Julius his best friend. He works for the Clockmaker by gathering people's clocks as a "side job," aside from his "real" job as a knight. Ace claims Julius is useless without him. Julius is one of two people whom Ace will allow himself to be truly defenseless around, the other person being Alice.
They do not get along. Because Gray is the other best fighter in Wonderland, Ace constantly wants to fight him even when Gray does not wish to, and enjoys irritating the other, especially to relieve his own frustrations. Gray considers him immature, while Ace does not like that he "replaced" Julius to Alice, in Ace's mind at least.
While none of the Role Holders seem to like Joker, Ace has the best relationship with him. As a means of trying to distance himself from his role, Ace helps out Joker in the prison. Even Joker seems to be annoyed at Ace's tendency to be extremely late.
In ~Mirror World~, Ace lives at the Castle of Diamonds. Crysta he considers a younger playmate, and they two will sneak out of the Castle together. For the most part they get along, with the main source of tension being that Ace sometimes tries to kill Sidney, which Crysta does not like.
Because in ~Mirror World~ Ace lives at the Castle of Diamonds, he is under Sidney's watchful eye, particularly when it comes to interacting with Crysta or when the Caucus Games are near. They strongly dislike one another and get into fights at times, especially when Sidney comes to the Art Museum to retrieve Ace when the boy visits Julius. Mostly Ace deals with it and Sidney tries to keep his temper under control.