Here's how I handle Capra Demon:
1. When you run in, go towards Capra Demon until you see him wind up for his attack. The goal is to bait him into swinging so that he doesn't run you against the wall like he normally does and give you nowhere to run. When you walk up close enough, he'll go for it (most likely the jumping double blade smash attack).
2. Roll past him and try to make your way to the stairs. If you get a chance to kill the dogs before then, do it. If not, wait until you get to the stairs then kill the dogs. The dogs have to go, they're the most annoying part of the fight.
3. Once the dogs are down, you can bait Capra Demon into attacking and punish. Step forward until he goes to swing, back up then tag him. He's wide open after the jumping attack, and with the dogs dead you have plenty of room to avoid it. If he does the normal slash attack, be sure to wait until the second swing. If you have Pyromancy or Sorcery you can get on the stairs and blast him, then roll off (you have to be at the top of the stairs to roll off, otherwise an invisible collision barrier stops you) and repeat.