As you explore Old Yharnam, you'll reach a valley containing an old church. Inside you'll find the vicious Blood-starved Beast.
Much like his cousin, the Cleric Beast, the Blood-starved Beast can be taken down without too much trouble as long as you don't get greedy. Also, you must bring Antidote to this fight, as the Beast can poison you with various attacks. (You can find some during the battle, behind the statue in the very back of the church.)
The basic pattern is this: sidestep his swiping attacks, chip away a little HP, and pull back. Use this strategy even after he's powered up - the only real difference is that he'll deal additional poison damage. He may also go into a rage and unleash a fury of attacks, so keep back until he calms down. Finally, try to stay locked on at a short to medium distance from the Beast throughout the fight; his lunging attack will send him off screen, out of your lock-on range.