You can purchase Lichtenberg figures made using a linear accelerator discharge into lucite. You could, in theory, make your own lucite artwork. The lucite (acrylic) is pretty easy to obtain. It's possible, though not particularly practical or advisable, to make your own DC accelerator. It may be possible to perform a modern-day version of Lichtenberg's experiments, except using polythene sheets which have been separated from each other and sprinkled with Lycopodium powder or perhaps very fine talcum powder. However, the method I think the average person is most likely to be able to attempt is to wood-burn a fractal pattern into pine. This can be achieved by applying 2-10 kV of voltage to a pair of nails which have been driven into a piece of dampened pine wood. In the example shown on the Tesla Downunder website, the author says he would spritz the wood with water if the burning process ceased (or presumably if the pine was in danger of catching fire). If you try to make a Lichtenberg figure at home, use care, and don't electrocute or burn yourself or others. Otherwise, have fun! I'd be happy to post photos of any attempts you make.
Lichtenberg Figures Website