they are pretty neat, i like the idea of the picture flowing across the nails.
i think the easiest way to do this would be to buy some very thin nail art brushes from a beauty supply like sally's, get the right color polishes. If you get fake nails you'll need the "hand" or something to hold them still while you paint them.
Get the picture you want to use and practice on a piece of paper with a pencil to get the basic "flow" of the picture.. then on your nails apply a basecoat, then it looks like she used a blue background, green, white and diluted yellow(with white) and black to paint. I don't think the design itself would be too hard, just getting the polishes to blend would be the most difficult. I'm pretty sure the artist mixed together the blue and green, green and black then white and yellow, so make sure you have something to mix a little of the polishes in.
i think the in the order of the colors applied was the blue, then the green octopus and treasure chest (maybe a mix of black and green polish), then the pale yellow accents, then the white highlighting on top of the yellow and finally the black on the far left octopus .
if you know who painted these, you could try to contact them and ask how it was done. it would probably be time consuming, but i like the design and concept.