This is the easiest of the breeding achievements. You will need to buy:
1 large chicken coop.
Some sort of trailer to move wheat or barley to the chickens.
A tractor with a front loader shovel (or not - see below)
First up, buy the coop - then go buy 100 each of the different colors of chickens (it does matter, the game counts each color as distinct for breeding, this will make it go faster)
Next, load up the trailer with wheat or barley - you can grow it or buy it at the store - go to the feed trigger on the coop and dump it in. Park there, as you might have to refill a few times.
Chickens breed when they are fed and clean - to keep cleanliness up, you have 2 options:
A.) the right way - use a tractor with a front loader to scoop up the grain that will appear on top of the feed trigger at least once per game day - you can dump it back into the feed bin.
B.) the cheaty way - this works for all animal pens - if you spread a thin layer of weed chips all over the feed trigger area, the chickens will not get dirty - if you are going with this method, it's best to do it in either 1x real time as soon as possible, or before you actually buy the chickens.
Now, just crank up the game speed, or go do some other work and your chickens will breed. Note that in the base game, you do not need a rooster for chickens to breed - if you are boosting this achievement only, do not use the seasons mod - it changes the mechanics a great deal, making this somewhat harder and much more time consuming. If your egg spawn trigger gets too full, things may slow down, so use your tractor to move the egg pallets off of it occasionally - you don't have to take them anywhere, just out of the way.