The drawing is called へのへのもへじ (so じ not し, mind the dots).
I guess this much is all most Japanese know (including me). The rest is what Google taught me.
As such, there is no conventions or principles on how letters are used for drawing. Using them liberally, apparently almost anything can be drawn.
There are variations to へのへのもへじ(see this for example). According to the wikipedia article above, there are region dependence and some other variations to へのへのもへじ are more common in Western Japan.
There is even a cat version. See this for a detailed instruction. (The picture is taken from the link in this page.)
So people are still drawing by letters, but not that anything is comparable to the original へのへのもへじ.
(source: 1, 2)
First, all above are called 文字絵 rather than 絵文字 (which reminds of something like 😊 or (^_^) which dates back to 1986-06-02.).
In Heian-period (794 - 1185), there was a type of drawing called 葦手絵, where letters were used for drawing reeds. There is no drawing left from the period, but modern reproductions look like this.
Later Buddhist sutras were used for drawing - but the source seems missing most images.
The origin of へのへのもへじ dates to around 17th century or later, mid Edo-period. Hiroshige drew へへののもへいじ like below （read top-to-bottom, right-to-left):
Hokusai used hiragana in his how-to book. This isn't exactly drawing by letters (more of making a sketch).
There is a type of songs called 絵描き歌 (drawing songs). This is not drawing by letters, but closer to what A.Ellett mentions in the comment. This is one of the most popular, and also many people should have seen how to draw Doraemon.