Bird killing can be done in a number of ways. You can shoot them down with arrows as they fly around your raft, you can stab or shoot them while they are pecking at your farms, or you can make a scarecrow and stab or shoot them while they are pecking at it.
However, the easiest way to kill birds is to craft a birds nest and place 10 - 20 of them on your raft. This will require 6 palm leaves, 6 planks, and 1 giant clam per nest. When you place the birds nest on your raft, birds will be attracted to it. They will leave feathers inside of the nests and will sit in them for some time. While they are sitting there you can either rush them down and hit them once with a metal spear or you can kill them with one metal arrow from a bow. The bow method is much easier as, when you get close to the birds, they will fly away and it can be difficult to hit their hitbox with the spear while they are in flight.