A look at how their relationship differs from the past incarnation and how this affects Roy in the present.
Roy states that he wants attention more than anything else which often leads him into trouble. His hacking into Oliver Queens' network lands him in jail but also gets him a job. At first things seem to be going well as Ollie gives him all the space he needs and acts like a buddy more than an employer. Yet something is missing and Roy can't find it in himself to voice this out loud. He tells Ollie he's happy but starts to have drinking problems. Once Roy discovers Ollie is actually Green Arrow things go on a downward spiral. Not only was he left in the dark but Roy felt he was being used. That maybe everything really was a lie. The two get into a verbal and physical fight where Roy is kicked to the curb.
According to Waylon Jones (Killer Croc) everyone knows that Ollie took all of Roy's inventions and shares in the company leaving him with nothing. Roy does take on the title of Arsenal in spite of Ollie to join up with other heroes his age. (*1) Not long things took a turn for the worst as he drank more and got into a fight with Killer Croc in the hopes of "suicide by Croc." Waylon picked up on this and refused to be used. Feeling for Roy he took him to get patched up, gave him a lair to stay in and helped him get sober. This had a profound affect on the archer and is in fact his most cherished memory. Ollie kicked him out despite acting as his friend and knowing Roy had been an alcoholic. Any support was offered came was far too late. Apparently no one in the superhero community cared enough to offer help. (*2) Yet a supervillain actually takes care of him and worries about Roy's safety.
This has instilled a sense of loyalty towards the scaly criminal who seems to be Roys' only confidant prior to the first issue. During his stay as prisoner Roy isn't his usual sunny self and it's easy to see why. He's going to be executed and doesn't think anyone will come to save him. As far as he knows no one cares about him. Only Jason and Kori do come to his rescue which leads to him being overly clingy especially with Jason, the most reluctant of the two. We learn that Jason and Roy have met prior to the jail break. (*3) They likely weren't close although Roy believes their make a good team despite Jasons' tough façade.
There is a big difference from Jason Todd's relationship with his mentor as Bruce Wayne is distant and isn't as easy going as Ollie is. Jason's more like a son while Roy is seen as an employee/friend. Bruce also told Jason the truth about his costumed identity and offered him a spot as his sidekick. Ollie keeps Roy in the dark and only mentions a partnership (*4) to calm him once things have become strained. When anything bat related is involved Roy expects Jason to want nothing to do with it until #17. Partly because of assumptions he's made on Jason's character but it's likely that he thought their experiences with their mentors were the same. Where Jason learns he needs to let go of his past to move forward Roy holds on to force himself to be better. Since Ollie isn't Roy's adopted father he doesn't try to take responsibility for his mistakes (or at least admit to them) like he used to prior to new 52.
Kori and Jason are the people Roy cares for the most. He's had to fight his way past their emotional barriers to earn their trust. What he has with them is what he's always wanted. While they were a little short with him at the beginning they never really pushed him away. In fact he actually hangs out with them on the beach and trails after them for missions so he's not left behind. Roy sees that they all have experienced similar things which makes them perfect companions. When he was working for Ollie he kept his mouth shut to pretend he was happy. With his friends Roy has no filter and tries to find out as much as he can. Being not just supportive but showing what he can bring to the group. To ensure that there's no misunderstandings on where they all stand like there was with Ollie.
Roy doesn't try to make assumptions where he stands with Kori until their both comfortable with their relationship. He does crowd Jason's personal space to test his limits and marks it as a win when he's not rebuffed physically even if he is verbally. There's some insecurity as Roy does keep tabs of Jason whereabouts and perhaps Kori in RHATO (#15) not a surprise as he has a few trust issues. He does succeed at getting both of his friends to open up and rely on each other in battle. Ollie still cares for him although it may take awhile before their relationship fully heals.
*1 Back in issue 6 it was revealed via Kori's memories that at the very least Dick, Roy and her used to work together. Lobdell intended for this to be the Teen Titans before this was vetoed. In a recent Secret Origins issue (the second telling of Dick Grayson's story) this was confirmed although their still not the TT.
*2 This is alluded to in #2 when Roy mentions no one returns his messages. How Dick and Roy get along hasn't been shown. Lobdell has said that he sees Dick as too nice a person to abandon his friend. Curiously Tynions' unprinted issue of Nightwing has artwork of Dick trying to be there for Roy during this time. It's odd because the work that Tynion has printed (#20) has a young Jason Todd saying Dick was friends with Roy before he fell off the wagon. This implies they weren't friends afterward.
*3 Tynion wrote the two meeting for the first time the day before Jason died when he was still Robin. This doesn't work too well with Lobdells' story as Jason was not only eager to be Roys' friend and goes against his own tale of Jason prior to his death. It's also confusing since Roy has no idea Jason cares about others in the first arc when Jasons' far more open with his emotions under Tynion. Lobdell seemed to imply that the two at least met as Arsenal and Red Hood since Jason told him that Roy that he was on his own some time before his capture.
*4 Under Tynion Roy knew about Ollie being Green Arrow from the start and tried to become his sidekick. Ollie was shown as more concerned for his well being and refused. This origin is retconned as Roy not knowing until a week or so before he gets kicked out. They were never officially crime fighting partners in either version although Lobdells' inspires Roy to become one to prove himself.