I wouldn't advise trying to follow the "Outline of the construction" section in the Wikipedia article. It it pretty abstract and not particularly enlightening, as it punts most of the interesting parts to theorems that it merely asserts or assumes known -- and most importantly it does not correspond to the more intuitively accessible Peano or Hilbert solutions shown in the figures. (So you're bound to become extremely confused if you attempt to match them up. They don't match).
Instead, what you could do is just take one of the standard curves that fill the unit square, and compose it with something like
$$ \langle x,y\rangle \mapsto \langle (1-y)x,y\rangle $$
that maps the unit square continuously onto your desired shape.