Like in the comics, his origin is a mystery. We are never given anything concrete. He tells different stories of what might have happened to him, that made him who he is today (just like in TKJ where he says that he prefers his origin to be multiple choice). The tales that he tells in TDK are also quite tragic, just like the story that was presented in TKJ of his pregnant wife dying and him falling into acid. We also do not know his name in TDK and that is how it is in most comics.
Like in TKJ, it is shown that him getting permanently disfigured is said to have played a big part in making him who he is. It is the same in TDK. In TKJ, him falling into the acid leaves him permanently looking like a clown. In TDK, he is left with scars that make it look like he is always smiling.
His belief that everyone is bad like him, once things become difficult for them. That is what he is trying to prove in TKJ with Gordon by crippling his daughter and showing him naked pictures of her and in TDK he tries to do the same with Dent (and succeeds), by killing Rachel.
TKJ "All it takes is one bad day to reduce the sanest man alive to lunacy. That's how far the world is from where I am. Just one bad day. You had a bad day once. Am I right? You had a bad day and everything changed"
TDK "As you know, madness is like gravity, all it takes is a little push"
His obsession with Batman and the way they were presented as the opposite of each other. He doesn't view Batman as an enemy, but rather as a challenge and as fun. That is how it is in most comics.
The way he caused everyone to leave the city, just like in The Man Who Laughs.
His use of Joker cards. The first time that the Joker is mentioned in the comics, a playing card is shown. It is the same in BB.
There are other ways that he is similar to the comics, but I am too lazy to write them all out atm lol.