If you have access to a vacuum you can put the blank in and the dye will be pulled deep into the wood. If you don't, you will probably be much happier with the results of applying the dye when you are closer to being ready to finish.
Turn it, sand it and then dye it with food coloring. I apply a CA finish with good results. A gel food coloring works well if you can find it. Sugarcraft makes a good one. http://www.sugarcraft.com/
The Americolor works well.