|Staff of air for Magic or Chargebow for Ranged.|
900 × Raw chicken (659,700)
3,461.58 × Feather (38,077.38)
720 × Egg (383,040)
The profit rate assumes 900 kills per hour. Your actual profits may be higher or lower depending on your speed.
Chickens can be killed at Fred the Farmer's farm, just north of the Lumbridge Market and its bank chest. It's important to use area loot for this method. Chickens drops raw chicken, feathers, and bones. The bones should be bound to your action bar to quickly bury it for prayer experience. The other items should be banked and then sold. There are also 3 egg spawns in the area, and chickens can lay eggs, which can be sold on the Grand Exchange for 532 coins, and raw chicken can be sold for 733 coins. Feathers are dropped in quantities of 5, 10, or 15 at a time and sells for 11 each.
As chickens have very few life points, a fast hitting weapon such as a scimitar is recommended for melee. For magic it's recommended to use a staff of air for an unlimited supply of air runes; abilities like Dragon Breath allow you to kill several chickens at once. One run could be completed in a minute with intense focus. There are 8 chicken spawns in the coop; if you kill all of them fast enough you have a few seconds before you can kill the second wave, after which you should have enough eggs and raw chicken to bank (the ratio may vary, this assumes 12 eggs to 15 raw chickens).
Close the door to Fred's house to prevent chickens from wandering in, making it harder to kill them and collect drops (being inside also causes you to pick up unwanted shears by area loot). Consider closing the gate as well when you are leaving for the bank to prevent them from wandering outside. All the chickens always respawn in the coop area.