I found someone used the recycling skateboard for making great creations. So just what kind of oil or paint I can use for dyeing on hardwood inside deeply(for laminating)!? Not just coated on the surface.
The handle in your photo is laminated, each colored strip was individually dyed before the lamination process.
It's been 40 years since I dyed wood but the process worked well. I made my own purfling and binding for my guitars. I believe the dyes I used analine dyes in powder form that dissolves in water. The other thing I did was to soak overnight in very hot water. I had a galvinized trough that I used for this. The wood I used for this was maple, birch would be good as well. You want a wood without open pores.
This can be a messy process, especially if you want several colors. You might also check to see whether what you want is commercially available. One place to look are luthier suppliers such as LMI and StewMac.
Wood dyes can be very expensive or very reasonable. Check out suppliers such Transfast wood dyes which come in a powdered form and can be tinted darker by adding more dye to the liquid base. You can use water but that will often raise the grain. I use rubbing alcohol from the big box which does not affect the grain. It goes on fairly evenly, dries fast and does not raise the grain. Other answers point out that your example uses veneer layers which are thin and can absorb the dye through the entire depth of the layer. IF you use thicker pieces, the dye will not penetrate all the way through. You may have to run a few experiments including rubbing on the dye and fully soaking the wood to find out the best technique for full color penetration.
Well that is not dyed hardwood, it is laminated (layered) pieces of wood which are brightly colored individually then glued together. Afterwards you sand or plane the surface and get the contrast like that.
You can also buy blocks which are already brightly colored. I think usually people will turn them. Sometimes they put layers of acrylic in as well to add more bold colors.