(Ed might be based in Gotham FX, but I take influences from various media & my own headcanons.)

Edward was born in Chicago in 1988. His intelligence and idiocrasies only angered his abusive father further.  Puzzles, riddles, were always a good escapism for him. When he was 17 he moved out, went to a university in Gotham, even though with his intellect, he could have gone anywhere. Why didn't he? Why did he choose Gotham? Maybe he will tell you one day.

His first 8 years in Gotham, he spent trying to fit in, trying to help, to make people like him. But no matter what he did, people thought he was weird and he felt alienated. Until he killed a cop, someone that hurt the girl he thought he liked. It gave him a power he never thought he had: stop the people that hurt him. When he saw he could use his intelligence to his advantage, he felt more like himself than he had in years. Still, something was missing. 

Meeting Oswald changed him. He found a friend in him, a guide. Someone he actually loved. He started committing more crimes, some of which he never told anyone about (including Oswald): he framed or killed abusive and corrupt people of power. Not many, not yet. That would come later, when he realised that cops and later the Bat, only caught him when he left them clues. When he realised how being underestimated, still, holds much more power than having obsessive fans that would do anything for you. But that would come later. Now, he is setting all logic aside in the name of love, but oh, he is not ready to accept that. He will project those feelings for someone else. This will destroy him in ways he never thought possible. Who would think that  betrayal and love would shape a  man that thought was ruled by logic?

After losing Oswald, Ed felt more lost than he had ever been. Lost, hurt, alone. It was the first time he had lost someone he loved (lost by betrayal, by death). It was the first time that he had no direction and everything felt meaningless. Someone saw him. A new him, still unformed, appeared. When Oswald came back, he found a purpose again, a game he could play (yes, that was the only reason he was glad, of course. how is it logical, after all to still want around someone that hurt you?). A game he lost because of his compulsion. A game that showed his weaknesses, weaknesses he would try his best to fight throughout the years.

Strange how being in good terms with Oswald, help him grow. After doing one of the less logical things he could have done (getting tortured, risking his life to protect Oswald) and yet being something he will never regret, he found a way to accept himself. To finally be the person, the villain he wants to be: someone smarter than anyone else (still driven by boredom and need for stimulation), someone that has fun with small games, someone that targets the corrupt & abusive (&few idiots), but also someone that that will let people underestimate him, someone that will stay in the shadows, patient, planning a grand finale. For now.

• IMPORTANT NOTES ABOUT HIS DISSOCIATIVE IDENTITY DISORDER: please do read THIS document. Note that I don't have DID, so feel free to let  me know if I am doing something wrong. Additionally, let me know if you are not comfortable with having this in our threads.

• HOW HE SEEMS  TO OTHERS: Even though there is a lot going on with him emotionally (accepting his sexuality, understanding his own emotions about people and situations, dealing with his mental illness etc), he is very good at pretending. His theatricality, his ever shifting voice, his love for games and riddles, give off a persona that says "I love having fun! And that's all there is". He does not want to show vulnerability to anyone and for anything (even to the people he cares about). If your muse doesn't know Ed, then they will probably see just that. 

• HOW HE TREATS OTHERS: After his first two kills, Ed started treating people differently. From wanting to help, he knows tries to learn how people behave to use it to his advantage. In other words, he does manipulate most people and he has hard time connecting with them. He will not betray your muse, not unless he is taking revenge, but he will not go out of his way to help them either. He prefers working solo anyway. But if Ed does care about your muse...read the next part...

• HOW HE TREATS PEOPLE HE LOVES: Ed does not love people easily (whether it is platonic or romantic). But when he does love, he often loses himself in it, especially at first, before he understands those feelings. He is willing to do anything for that person, to lose everything, even his life. He wants to prove himself to them, over and over again.

• BEING LOVED: Ed doesn't believe people can actually love him. They may admire his intellect, they might like his games, they might even find him useful, but love him? He doesn't think it can happen. And even if it does, he thinks that the moment he makes a mistakes it all stops (see: how he surprised he was when Oswald rescued in 4x15).

• MORALITY: Despite his 'villain' persona, I think he is morally grey. He does not mind killing people, but he does not kill just anyone. He kills those that he deems to be corrupt, abusive, those that have hurt him & people he cares about. He believes that those people must pay and since the cops or the bat aren't going to do it, he might intervene. He is using those people for his games, of course. Sometimes those games become known, leaving his question marks and riddles. Other times, he puts them in escape rooms he has made. Others...he keeps those crimes to himself. That being said, he is not the one to rescue people or help anyone in need.

• HE IS DRIVEN BY: boredom, need for mental stimulation, curiosity, compulsion (when he formulates a plan, it has to happen), love

• THEATRICALITY: especially as the Riddler, Ed tends to exaggerate his mannerisms. He uses his hands a lot to make a point, he uses his hands and occasionally a cane as props to add to it. Being more reserved is an indicator that he might know your muse well...or that he is not in a good place