A Basic Chicken Farm
Build a pen
Dig a hole, 5 x 5 x 1 block deep.
Place fences all around the edge.
Put a block in the middle.
Fences form a barrier that is 1½ blocks high, so that mobs (or players) can't jump over it.
If you place a block in the centre of the pen, you will be able to stand on it to jump out of the pen, but the chickens won't.
This pen design is easy for you and the chickens to get into from any direction, but only you can get out.
There are two ways to add chickens to your pen:
Each chicken will lay an egg on average every 7 minutes. In areas where there are wild chickens, you'll find plenty of eggs. If you have 20 or so, that's enough for a good start.
Just right-click to throw eggs into the pen. They will break, and one out of every eight will hatch into a chicken.
First you need some seeds: you can get these by breaking long grass, or if you are lucky enough to find a village with wheat growing, you will get seeds when you harvest that.
Find some chickens, and if you are holding seeds in your hand, they will follow you wherever you go. As long as you go really slowly, they don't like to rush.
Once you get the chickens home, just go into the pen and they will follow you in. They will start laying eggs, which you can collect to make more chickens with. Once laid, eggs will vanish after 5 minutes, so collect regularly.
Breed more chickens
If you have seeds from grass or wheat, you can use it to make the chickens breed. Right click on them, and hearts will appear above them. For every two chickens you feed, a baby will be born.
Now for the sad part
At a certain point you're going to have to kill your chickens. Just jump in the pen with your sword out and start hacking. Make sure that you have enough eggs to start over before you go and kill them all!
Cook the chicken and make arrows with the feathers.