The professors who found this actually wrote a paper [1] explaining every step that they did to create their supercapacitor, including the formula for the gel-like electrolyte they used. The biggest problem I've found to making this at home is getting graphite oxide. They are expensive on the net (google it, not linking any store because I don't know any trustworthy), or hard to make (the usual method requires strong oxiders like sulfuric acid) [2]. There is a bottom-up approach to making them, that uses glucose [3], but I have no idea how hard it is, I need help from someone more knowledgeable in Chemistry before I try it. So if you want to really try this out, read those papers.

[1] http://www.sciencemag.org/content/335/6074/1326.abstract [2] https://en.wikipedia.org/wiki/Graphene_oxide [3] http://pubs.rsc.org/en/Content/ArticleLanding/2012/JM/c2jm15...