He took me a few tries on my first playthrough until I managed to kill him and every playthrough since then I've been able to take him down first try. I think the difficulty with the Capra Demon comes from the sheer element of surprise he has, and you are given very little room and time to try and develop strategies to fight him.
The strategy that seems to work very well for me is as soon as the fight starts, sprint for the top of the stairs, rolling to avoid attacks on the way, then when at the top of the stairs kill both dogs as fast as possible. If you successfully managed to kill the dogs then the fight is a breeze as long as you are quick at dodging the demons attacks. Some of his attacks can break right through your shields so if you aren't familiar with his moveset then you are better off just dodging every attack he makes.
The hardest aspect of the fight is the lack of room in the arena you have to fight the demon in, you can solve this problem by kiting him quite easily by leading him up to the top of the stairs and then jumping off the stairs, he will do the same and you can lead him back up the stairs in a never ending cycle.[footnote]Well you take a tiny amount of fall damage but you can kite him for a very long time. [/footnote] Any ranged attacks make the fight even easier.
EDIT: Oh yeah, since you started as a thief, you will have a master key, with the master key the capra demon, the entire depths area as well as the annoying part of Blighttown are all completely optional so if you're really having a lot of trouble with him you could just skip the fight entirely and maybe complete it later on when you are stronger. You will need to kill the Capra Demon to get the large ember though. I highly recommend skipping the annoying Blighttown section though.
The strategy that seems to work very well for me is as soon as the fight starts, sprint for the top of the stairs, rolling to avoid attacks on the way, then when at the top of the stairs kill both dogs as fast as possible. If you successfully managed to kill the dogs then the fight is a breeze as long as you are quick at dodging the demons attacks. Some of his attacks can break right through your shields so if you aren't familiar with his moveset then you are better off just dodging every attack he makes.
The hardest aspect of the fight is the lack of room in the arena you have to fight the demon in, you can solve this problem by kiting him quite easily by leading him up to the top of the stairs and then jumping off the stairs, he will do the same and you can lead him back up the stairs in a never ending cycle.[footnote]Well you take a tiny amount of fall damage but you can kite him for a very long time. [/footnote] Any ranged attacks make the fight even easier.
EDIT: Oh yeah, since you started as a thief, you will have a master key, with the master key the capra demon, the entire depths area as well as the annoying part of Blighttown are all completely optional so if you're really having a lot of trouble with him you could just skip the fight entirely and maybe complete it later on when you are stronger. You will need to kill the Capra Demon to get the large ember though. I highly recommend skipping the annoying Blighttown section though.