Bloodborne

The Cleric Beast trophy is awarded for killing the Cleric Beast and will likely be the first trophy you obtain in Bloodborne. The beast itself can be found on the Great Bridge in Central Yharnam and is the first boss fight in the game. Yharnam can be a bit of a maze, especially for the uninitiated, so I'll provide some directions for you.

Directions
To reach the boss, start at the Central Yharnam lantern. Head down the right hand set of stairs in front of you. As you cross a small bridge you will see a street down below and a mob of hunters and townsfolk ambling further into the city. Make your way down to this street and follow the mob, clearing them out as you go. As you round the bend, you will see a massive bonfire, surrounded by more hunters. You will need to clear them all out or make a mad dash past them, up the left hand set of steps and through a door into the courtyard beyond.

On the other side of the courtyard is a set of steps. Make your way up these steps, clearing out a pair of dogs and their attendant hunters. One more flight of steps gets you to one end of the Great Bridge. To your right you should see a pair of beasts blocking your path. If your skilled enough, take these two on and clear the way forward. The Cleric Beast will be at the far end of this bridge (see Boss Fight further down).

If you don't want to fight the beasts, there are some alternatives. Return to the courtyard and look along the fence for some destructable debris. It conceals a gap in the fence that will let you drop down to a lower road. Some mangy dogs guard a group of dog cages but past them is a route passing under the Great Bridge. You'll enter a dark house; the denizens insides will attack you if you get too close. If you take the stairs up and out of the house, you'll end up on the Great Bridge, past those two beasts from before. If you take the first floor exit and continue through the house, you'll open up a shortcut back to the Central Yharnam lantern.

There are additional paths through the streets that lead to the Central Yharnam sewers but since these are longer and more involved, I'll leave them for you to discover.

Boss Fight
Once you make it to the end of the Great Bridge, the Cleric Beast will climb over the far wall and drop down to fight you. Once a hunter and cleric of the Healing Church, the Beast is now lost to the curse and cannot be tamed.

He'll attack you with a variety of moves, usually sweeping at you with his long left arm or smashing his fists into the ground in front of him. The key to the fight is close quarters combat; you're firearm won't do any damage from a distance and if you get too far away, the Beast will leap into the air and smash down in front of him with a decent ranged attack.

Your lock-on on the boss can be helpful; it makes circling around him much easier and keeps you on target when he looms above you. However it's easy to lose track of your surroundings while locked on, often leading to you getting cornered or trapped on an obstacle followed by furious smashing from the Beast. Use it as you see fit. You should be able to dodge between the Beast's legs as he attacks you. Staying behind the Beast keeps you out of range of his melee attacks and gives you some time to do damage. If you're not fast enough when dodging his close up grabs, he'll haul you into the air and smash you down fast, dealing massive damage.

Keep dodging and slashing at him as often as possible. Back off and use a Blood Vial if you need too. Be swift in attacking him after he's done damage to you; you'll get back some of your health if you can do damage to him in return. If you do die, you can return to the Hunter's Dream and talk to the Doll, who will help you use your Blood Echoes to level up. If you find any Blood Stone Shards while exploring Yharnam, you can use them to level up your weapons at the Workshop in Hunter's Dream too.

For those who like a visual guide to the boss fight, check out the following video from VaatiVidya:
If you are victorious, the Cleric Beast will drop the Sword Hunter Badge and will give you access to the Great Bridge lantern. Congratulations!