Butcher blocks can be made from any wood although I would recommend a hardwood. Oak, walnut, or maple come to mind. Oak has a pronounced grain and the wood will express more visually. Maple or walnut are more closed grain and should give a more uniform appearance.
I can't speak to the permanence and uniformity of ink, so I would recommend using a dye such as TransFast black powdered dye. There are other dye colors worthy of consideration as well. Dissolve it in denatured alcohol. Water can raise the grain requiring additional sanding which, of course, can re-expose natural wood color, where alcohol will not. Use a test board to determine how concentrated you want the mix and how many coats give you an even finish. The pigments in dyes are much smaller than in stains and allow the natural beauty of the wood grain to be expressed while achieving a deep rich color, where stains actually can hide the grain more.
Your 220 grit sandpaper should be adequate, although you may want to go for a final of 320 and steel wool to finish it off. Use a sanding block or powered sander to insure a flat surface is maintained. If you have access to a paint sprayer you can achieve a better result than with a brush. Although a mix of 1/3 each of linseed oil,, varnish and mineral spirits can be applied with a rag (it may take 3-6 coats to achieve your final finished appearance).