The best way to enrich your yard organically is through natural compost and manure. Chicken manure is excellent (cow and horse manure may contain seeds of unwanted weeds). Another great thing to do is to start a compost pile and spread a thin layer over your yard once a year.
You can now buy organic fertilizer at nursery or home supply store. It consists of composted organic materials. Find the fertilizer ratio (N, P, & K) before using it.
There are certain times of the year that you need to fertilize your grass. If you do it more then you should you have a great chance of destroying your lawn. Organic products are just as harmful is they are used incorrectly. Talk to an expert or read up on this issue before you start anything.