I have done this with a guitar with light rosewood fretboard and had really good results - the board turned very black and wouldn't come off on the fingers at all.
It's not hard to do, but preparation is the key. You need to really, really clean the fretboard. Start out by rubbing it down with a cloth/toothbrush and naptha. Once it looks super clean, very lightly sand with some 400/600 grit sandpaper. Then I'd go over it one more time with the naptha. Board should be bone dry.
Go to an arts and craft store and buy some of this stuff (it's only a few dollars):
Tape off anything you don't want to get ink on with some painters masking tape. Get a fine paintbrush, and paint the ink onto the fretboard (try to avoid your fretboard inlays - they will darken slightly if you get stuff on them). Once you're done the whole board, let it sit overnight. It will dry and soak in. The next day, repeat (you will see any minor spots you missed). Then after waiting another night for it to dry completely and soak in gently rubbed it down with a rag to remove any remaining black stuff from the surface, then did applied a light coat of Watco Danish oil and gave it another night to soak in/dry.
After that it should be playable. Rub the board down with a rag, and you shouldn't have any black coming off it. If you do, apply another light coat of Danish oil and wait another night for it to dry. Finally, take a sharp knife and gently/carefully scrape the top of your inlays clean anywhere you slopped ink on them, and you're good to go. Should be very black and unless you sand it down it will stay that way.