Sometimes things just don't go to plan, we've all been there with Capra - but failed attempts aside, running to the stairs to get some space and despatch the dogs is a sound strat.
A few things to try:
- Don't immediately run. Go through the fog and wait a beat, 8 times out of 10 Capra will do his leap attack (or just take a swing), roll through that and use the time of his recovery animation to get some distance or take out one of the dogs, then run up the stairs and deal with the other.
- I prefer to go to the right first around Capra. Enemies will follow your direction, so it should give you space to run to the stairs rather than trying to squeeze by Capra on the left and getting cornered.
- Spears and rapiers can attack from behind a shield - great for the dogs. Just watch your stamina as it regenerates far slower when your shield is raised (no stamina = guard broken, stun-locked).
- Weapon. You can have a +5 weapon by this point, if you don't, why not? More damage = quicker battle, less chance of dying.
- Explore down the stairs around the corner if you haven't already, I prefer the run back from Firelink rather than going through all of Lower Undead Burg from the Burg bonfire.
Mages can use hidden body to cheese the dogs. Wolf Ring can add poise to any build. And when all else fails throw shit (dung pies) over the wall for maximum cheese. Also Capra is technically an optional boss, there's certainly no reason to do him 'on schedule' (after Gargoyles), there's much to explore besides there, although the Depths (the area after Capra) has an important treasure.
Everyone has been nobbled by Capra at some point don't fret about it :)