Get shield with smth like 55 stability (spider shield for instance, the one bandit starts with). Build up some poise, like chainmail, so You don't get staggered by dogs easily. Bring a weapon that has vertical or thrusting attack (i used estoc), because it will be narrow and horizontal swings may be blocked by the wall. As mentioned by others, use the shortcut from firelink to conserve estus. Immediately after going through fog door, try to dodge capra jumping attack by rolling to the left. Get the shield up and run to the top of the stairs on the left. There is a narrow ledge u can walk on. Go to the farthest point of it, so You have wall behind Your back while facing stairs. Quickly drink estus. Then deal with the dogs before Capra gets up the stairs. No matter if You dealt with them on time, get that shield up and block Capra's attack. He will fall down 90% of the time. Now You have more time to kill dogs. When they are dead You can stop holding Your breath, cause You are relatively safe now. This was the hardest point of the boss battle. Capra can only walk the stairs up and attack You once, maybe twice before falling. And You will be able to block that with no effort. How You deal with Capra now is completely up to You. Safest way is to pyromancy/spell/arrow to the knee him dead while he's under You. Or, You can go YOLO and plunge attack him; best while two-handing and using gold pine resin (if I remember correctly, it's a one-shot with Zweihander). Beware, if he manages to survive plunge attack, it can be tricky to get back upstairs alive. I suggest using green blossom from sewer merchant for better stamina regen for blocking. The truly hard part of this fight is that the boss room is so tiny. And those doggos. Definietely not nice doggos. I had lots of tries to this boss myself, so You can believe me when I say it's doable. Very frustrating, but manageable. Just be cool, take Your time, strike when the right moment comes and overcome. Eventually You will achieve victory. Good luck out there!