I was actually looking at doing just this earlier today.
That being said, I need to spend a bit more time researching it as there does not appear to be any public api available for steemit, authenticated or otherwise. I could be wrong, so if you know where the api endpoint is, please let me know.
From what I could see, it looks like I'd have to grab a copy of steem source, upload it to my own server, create an authentication mechanism for users of the system, expose the rpc calls that I want to use via an api, and then write a plugin to take interface the api that I expose.
Anywise, I'm still looking at it, but with my limited free time I'm not certain that I've the available time to tackle all of that currently.
If there is a publicly available steemit api that I'm missing, please point me in the right direction and I'll start development asap ;)
Quick update: I've created a bridge plugin for wordpress that lets people showcase their steemit.com content on their wordpress site until such time as we get a public api or someone puts serious effort into doing what you note. See here: https://steemit.com/wordpress/@tony.jennings/show-your-steemit-posts-on-your-wordpress-site