First off, you need to understand what the difference is between these methods.
Organic- This is the process giving your plants the nutrients they need like nitrogen, potassium, etc, by simply adding natural nutrient rich compounds. Some examples are, different types of animal dung, worm castings, and bat guano. All of these different types of animal droppings have different nutrient values to benefit plants in different parts of their life cycle.
Pros: readily available, cheap, low risk of nutrient burn, forgiving, easy to get started, natural tasting finished products
Cons: Less potent nutrients, slow-medium acting, storage, not-discrete, potentially less effective.
Chemical- This method is usually achieved by adding a store bought chemical compound to the plant's water supply with the desired nutrient compound in order to benefit the plant.
Pros: Fast acting, strong, accurate nutrient content, potential for large gains
Cons: Expensive, Hard to find sometimes, Nutrient Burn, Hard for beginners, possible chemical taste in finished product
No-one can tell you which way is the "right" way to feed your plants nutrients, you just have to understand the alternatives. I will post 3 follow-up articles explaining how to add nutrients, and what nutrient burn is. Stay tuned!