Vendrick is the giant you walk past to get the King's Ring after defeating Velstadt The Royal Aegis in the Undead Crypt. Don't attempt to fight him though, beat the game first then come back.
(New Game + does not automatically start after beating the final boss)
Vendrick has very high defense so he will take some time to kill, additionally his attacks are likely to kill you in a single hit. However it is possible to lower his defense by having the unique items "Soul of a Giant" in your inventory (DO NOT use Soul of a Giant for the souls). Three Giant souls can be obtained from killing the giants in the giant memories and the one under Black Gulch.
For the fight I highly recommend a fully upgraded weapon and using a 100% block greatshield with high stability. A good shield will allow you to take a hit now and then. Surviving a hit is much better than the damage two handing gives you as this is very long fight.
For reference I used a lightning mastodon greatsword +10, with 40 strength 35 faith and was doing 260 damage a hit while two handing.
Patience is key for this fight so take your time and don't get greedy.
Once the fight starts sprint right to him (you might need to dodge through an attack) and hug the back of his left leg (opposite the side he holds his sword). This sweet spot is where you should stay for the entire fight, if you're lucky. If you strafe counterclockwise he shouldn't be able to hit you. Wait for him to attack then land a few hits and continue strafing. Keep in the middle of the area, stay calm and you should be fine.
In the event he jumps away or starts to hug a wall, sprint away, wait for an opening or for him to get into the middle, then sprint right back
The Achievement unlocks once you beat him.
If you want his soul and armor it's located behind a locked door a bit passed the second bonfire in Shrine of Amana. You have to have beaten him and be in human form for the door to open.