Anime characters have crazy hair colors and styles for a reason--the characters need to be easily distinguished from one another. So if you do your hair like anime characters' hair, the most important goal is to make it look different from everyone else's.
Pick a hair color. Colors like blue, green and flaming red come to mind, but most importantly, the color should be different--even outrageous.
Dye your hair in this color. This is one time when it doesn't matter if the dye comes out exactly true to color. If you're trying for red and it comes out purple, that may be even better!
Wash your hair and use a conditioner once the color is secure and you're ready to style it. You definitely want a leave-in conditioner that makes your hair thicker and fuller.
Decide on a hairstyle that works for you. If you are guy, try multiple large spikes or slick your hair straight in front and back. If you are a gal, style it into huge sweeping pigtails or a very thick bob.
Comb or brush your hair long enough to get it straightened out. Any tangles or knots in your hair will make the styling process more difficult.
Brush, tie or stretch your hair into the preferred look, using enough gel or hairspray to make it stay. Bows or ribbons will be needed for ponytails. For spikes, children's art glue works best because it can be washed out.
Choosing a specific anime character may work best because you'll have a model to follow.
Make sure your hair is long enough for the style you choose. Short hair won't work very well in getting the excessively long spikes, slicks or ponytails you're looking for unless you use extensions.