They were rivals in college. Both incredibly brilliant minds. Doom was interested in the supernatural, Reed thought that was hogwash.
Reed interrupted Doom's experiment wherein Doom was trying to travel to another dimension. Doom was trying to travel to Hell to battle the Devil for the soul of his mother. Reed's like "whatever bro, I'm an atheist, I'm just saying your machine is fucked". Doom is all "bullshit dude, you just don't understand MAGIC".
Problem is that Reed was right, Doom's machine was whack and literally blew up in his face. Doom makes Reed a scapegoat for his failure and becomes obsessed with proving himself superior. Which, fair, Reed's a bit of an egotistical prick himself and needs to be brought down a peg.
But that's just Earth-616.
The idea of Reed/Doom not being a thing in the MCU is the one thing that gives me MAJOR trepidation, but that didn't stop me from getting SUPER EXCITED after seeing this trailer.