Okay, first things first, find a colore. They look like the picture below.

Dig those things up and get "Dye Fossils". They look like this.

Next, you can either make a "Portable Dye Extractor" or a "Dye Extractor". First we'll talk about the "Portable Dye Extractor. This is how you craft it.

Next, go ahead and take your "Dye Fossil" and place it in a crafting grid with a water bottle and violà! You have the dye that the fossil relates to!
Be careful though! The "Portable Dye Extractor" has a limited number of uses! Side Note: The recipes are shapeless.

Next onto the "Dye Extractor". This is a stationary block that will extract dyes from fossils without breaking.
First, craft a "Dye Extraction Kit", recipe below.

Now you can craft a "Dye Extractor". Recipe below.

Place it down, right click it and BAM! A whole new GUI.
But Chris, "What do do all the slots do?" I hear you ask. Well, look below and you'll see.

Dye in the left slot, fuel in the bottom. Fuel is either a Water Bucket (You get the bucket back), or a Water Bottle (You dont get the bottle back).
The Water Bucket also lasts twice as long.