You will learn about this quest by reading the note after interacting with notice board in Velen. Go to Crow's Perch and meet with Chet sitting in front of a hut (M3,4). You can immediately start searching for shrieker or first bargain for better payment for the task (you should be able to bargain about 270 crowns). Don't leave the castle yet, first go to the back of the hut. There you will find a boy named Symko. Talk to him about the beast you are about to hunt. You can immediately end the conversation or use Axii on the Symko (if your sign is developed enough). It will calm the boy and give you additional 25 experience points.
Leave the perch and go east. After reaching the marked territory, activate witcher senses. Find the dead animal and examine the large footprints of the shrieker on the ground. There is a blood to examine as well. Follow the monster's trail. You will get through a small brook and climb some rocks. Investigate the tracks near the cave twice (you will unlock a note about a shrieker in the bestiary). Go into the cave (25 experience points).
After entering the cave, Geralt will determine that the Shrieker is in an unavailable place and he can't fight him because of that. You must use bombs to make a path. The game will suggest three bombs - Samum, Grapeshot or Dancing Star. You should have a blueprint for at least one of those bombs so you just need to create it. Required alchemy ingredients can be either found or bought from local merchants.
Return to the cave after you obtain at least one required bomb. Go towards the noises of the beast (red "waves" visible after activating witcher senses), aim the bomb at them and throw it. If you do it correctly, the shrieker will leave the cave.
Important - before you start the battle, make sure you have enough healing items to regenerate your energy for a long time. If you have draconid oil schematics, put it on Geralt's silver sword (winning the battle without it is possible but it will be harder). Don't start the battle before you reach 7-8 experience level.
Leave the monster's nest and go to the nearby field above which he will be flying. This is where the proper fight with the shrieker will take place. This enemy behaves similarly to the griffin from White Orchard - he flies above the battlefield and from time to time lands to participate in a more direct combat with Geralt. If the shrieker is currently in the sky, try to observe his actions and jump away whenever he will try to attack you. It is a good opportunity to heal yourself (later in the battle) - his attacks aren't frequent. You don't need to force the shrieker to land, sooner or later he should do it anyway.
Attack the shrieker after he lands. Don't rush it, start by stunning him. Grapeshot bombs and Aard sign should help you with that. After you successfully stun the shrieker, run towards him, hit him with two-three fast attacks with a silver sword. It is VERY important to jump back after a series of hits and to focus on dodging the monster's attacks during close combat. His attacks are very strong, and they can additionally stun Geralt or activate bleeding status. If this happens, concentrate on regenerating and dodging. Resume attacking the enemy after Geralt regenerates and will be able to use a bomb or the Aard sign again. In the final part of the battle, shrieker will start running away. You shouldn't have trouble with catching the beast and dealing the finishing blow.
After defeating the shrieker, examine his corpse. You will obtain some valuable items, including Cockatrice Trophy. Now you can return to the quest giver and inform him that the monster is slain. Chet will want to give you gold. You can take the payment (the amount you have previously bargained + 235 experience points) or let the man keep his gold. In second situation you will receive 250 experience points.
Main reward for completing witcher contract: Cockatrice Trophy - 10% chance to cut off a limb.
Contract: The Griffin from the Highlands
This witcher contract becomes available later in the game - check the notice board in Crow's Perch regularly and during one of the visits in baron's quarters, the new quest will be added to your journal. The contract itself is very simple - you must go North from the Perch, to the territory terrorized by an archgriffin. Red marks (use witcher senses) will help you find the monster. Important - don't start this quest until you reach at least level 24-25 because the flying beast is on the 26th experience level.
After getting close to the archgriffin, start attacking it with the crossbow. At the same time, avoid its air attacks (jump to the sides). Similarly to when you fought with the regular griffin at the beginning of the game, you must force it to land. After that, start the real battle - stun the archgriffin with Aard sign, use the silver sword with hybrid oil and additionally throw Grapeshot bombs at the monster. After killing the beast, search its corpse, take the trophy (10% chance for cutting a limb) and report to the quartermaster in Crow's Perch. You will receive gold as a reward.
Important - Archgriffin drops Archgriffin Acid Gland as well. It can be used to complete the side quest Master Armorers so don't sell it.