These three phrases have different connotations:
Of the three, the last is most dissimilar to the other two. "I am interested in doing this" signals a willingness or availability for a task. For example, "Did you see the posting about volunteers wanted for the playground build? I am interested in doing this." The speaker does not necessarily have a literal "interest" in the task, but they have the desire to do it (for whatever reasons). This is idiomatic English.
The other two phrases are closer in meaning, though I might say that "this is interesting to do" implies a sense of activating one's imagination / intellect / curiosity. Whereas "doing this interests me" connotes that the speaker is engaged with the activity, though maybe not exactly as above. More like, this occupies my attention, rather than my imagination.