I want to use cactus Juice and black alumilite dye to color some wood that I want to stabilize. Do you have any recommendation about the concentration of the dye to be applied?

There is no way to even begin to guess at this without seeing and feeling the wood.  I would probably start with a ratio of 1/2 oz per gallon and run a test piece through completion.  Then cut that blank in half length wise and wet the surface to see the color.  If it is not dark enough, add more dye and repeat with a new piece of the same species.  If it is too dark, add un-dyed Juice and repeat.