Not much is known about her but what we have seen has been intriguing.
Essence was the youngest of a clan of cavemen that were turned into immortals known as "The Untitled." While her mother Durca attempted to rebel against the will of their leader Essence was a child at the time and didn't have any say in the matter. To maintain the peace Durca created the All Caste. Eventually a newly resurrected Jason Todd was brought to them to train in hopes of fighting off his own inner demons. Things didn't start off smoothly between Essence and him. Jason claimed he saw an emptiness in her and death would be a mercy. (*1) This detail does get my attention partly because her kind is called the Untitled and her choice of name seems to imply she wishes to fill the void within herself.
Jason and Essence do grow closer with Jason admitting to himself that she's his first kiss although he denies she's his "ex" in front of his friends. Just what they were hasn't been clarified just yet, we also don't know how she feels. It's fairly obvious that no matter what they were she means something to him. When she was exiled by the All Caste he vowed never to return and only went back she asked him to go after they died. Essence does seem to care but she's fairly manipulative to get what she wants. Not only does she use his feelings for her to get him to join her hunt she directs him while withholding the truth. Although Essence is following her mothers' footsteps in this regard since she also kept Jason in the dark.
During Jasons' confrontation she suggests she was only going to keep the Outlaws alive if they did what she wanted. Although she was furious at the time so that could have simply been her anger talking. It's unknown if Lobdell would have kept her the traitor of the All Caste but it's an intriguing twist. Her initial refusal to reveal herself to Kori and Roy could be seen as dismissive. It also gives them the impression Jason is talking to himself which she either didn't think they'd notice or didn't care.
It's noted that Essence has a bad temper and since few things have been able to harm her (*2) it makes her nearly unstoppable. Just having Kori dare to attack (which was only to protect Jason from her wrath) pisses her off to the point she takes it personally. The Untitled seem to see humans as inferiors and believe they should rule over them. Essence does admit that she can at least understand this thought process since mortals are highly frustrating and don't get the bigger picture. Since the Untitled literally have evil inside of themselves there seems to be an ongoing fight within. (*3)
Tynions' run happens and there's only a few things to take away from it in terms of her character. Essence wasn't responsible for the All Caste deaths and she's on good terms with Jason. (*4) When Roy is burnt and near death she heals him as a favor. In his narrative Jason says they will always be there for each other.
Perhaps they weren't entirely romantic during his days training with the All Caste, at least from her perceptive. He was still a kid (he might still be seen as one given her age) going through a devastating period in his life. Essence also has her own demons to slay and doesn't make time for him unless it's important. Just what she's been doing since she's been exiled (besides trying to kill other Untitled) isn't known. I do wonder if what's hinted at in Futures End is something she knows about. If Essence at least suspects Jason is/will become immortal and thinks she can wait for him. What if you found someone you liked who was younger and more immature but knew you potentially had all the time in the world? You wouldn't rush to be with them either.
Her distance might be partly due to the corruption she's battling. Something that might not just be within as her body--at least under Rocaforts' pen--seems to show physical damage. The veins on her forehead and peeking out on her neck. There's even stitching between her eyebrows. The most obvious sign of darkness being her entirely black eyes.
There's also something about her and her relationship with Jason that caught my attention. I'm not sure if Lobdell did it intentionally but a while back Jasons' death certificate was released. According to it the explosion and the beating he took didn't kill him. It was "asphyxiation due to smoke." What's Essences' main ability? Turning into black smoke. It certainly adds another twist given her immortality and how he could view her. Depending what Lobdell decides to do Essence could have a lot of interesting storylines.
*1 Secret Origins #5, it's unknown if Jason was projecting his own issues on Essence or really did see an emptiness in her. The former comes to mind because of Jason fearing himself after rising from the Lazarus Pit.
*2 At least under Lobdell, I have no idea why Kori is able to do anything against her under Tynion. Again I'm mostly ignoring what Tynion wrote since a lot of it doesn't work with what Lobdell has done with the character.
*3 They got the powers from the "Well of Sins", the All Blade only reacts to pure evil which was within Essence and she talks about the corruption in her back up story.
*4 She was also made more powerful and was putting a new All Caste together. It's unknown if she's still doing this. Essence implies she's busy although her clothes aren't white like they were when she gained her power.