To say "As a child, Did you/I read books for fun" I would use
子供の時、趣味で読書をしていましたか?
〜で is simpler and the sentence flows better than 〜のために, which can be used to mean the same thing. However, when I say
子供の時、趣味のために読書をしていましたか?
actually sounds a bit awkward, since 〜のために (or ため in general) literally translates to "for" but it has a tiny bit of connotation to indicate whatever is done is a requirement, or something purposefully done. The reason why the specific example above sound a bit awkward to me is that your hobby is not necessarily something you have to do; you naturally do it out of enjoyment. So using 〜のため to indicate your purpose, I think, has an effect of emphasizing your will to do that thing.
If instead the sentence is
昨日、仕事のために読書をしていましたか?
then the awkwardness goes away, since the reading was a requirement for the job, something needed to be done by will.