The key to the Batman VS Superman matchup is Batman's trait. Batman's trait is an ESSENTIAL (I can't stress that enough) in this match. The bats provide you a way to get in on Superman and nullify his jumping lasers. You must always have one available. When you do have at least one bat, release it as you see the start up for any form of laser. If you can catch Superman in the air doing the air laser, the bat will knock him down and you get free dash in damage as he falls.
Bats can help you when Superman starts the F23, breath pressure. If you have one stocked and you're blocking, you can release the bat in between block stun to knock him out of pressure. I've only done this a few times as the window is very tight.
Other than those tips, you just need to play fundamentally sound and block while making reads and picking your spots to attack. Batman's jump attack can be anti-aired easily by Superman with D2 or his rising grab. However, if you can condition your opponent to be expecting the J2 (or if he makes constant attempts to stop it) you can stop the dash in a bit short, defend his anti-air attempt, and hit with a low. I like to dash in, stop just outside of striking range, let his anti-air attack whiff then punish with B1 combo or slide.
As far as Doomsday goes, you need to play him like a more mobile version of Bane. Batman's dash is a good tool in this matchup. When he does the jumping off screen attack, dash forward and punish. You also need to know how to block the ground slam which is an overhead then low. Also, when Doomsday's trait is up, dash and jump away.
