I think he's trying to kill Shoto coz he doesn't want Enji's goal to be a success, which is producing a powerful quirk under his lineage or some shit.
I think Dabi resents Shoto because while Dabi was deemed a failure, Shoto was considered a success quirk-wise. Endeavor paid special attention to Shoto and tossed aside his other children. I imagine that to Dabi, a child at the time, it just looked like his brother was favored over him and so Dabi came to hate Shoto too.
Dabi 'died' when Shouto was five so he prolly didn't see the abuse Shouto went thro. He probably only saw his quirk manifest and Endeavor started treating him better
Dabi said he had planned to killing Shoto but changed his mind when Enji became number 1, but why would he want to kill Shoto? He went through the same thing as him, so wouldn’t you want to just kill Enji to get revenge on him for everyone?
Is he perhaps partly mad that Enji gave up on him?