G'day Matt. Welcome to the forums.

The Overland Track has a fairly steep hilly section near the beginning (up to Marions Lookout), and a few other hilly sections (but none as hard as the first one). For a lot of the way, it is reasonably flatish.
If you'd like to see a list of gear that I usually take walking, I've recently posted my list to another topic in these forums,
HERE. I hope this helps!
In June it will be very cold indeed, and you will probably see a bit of snow, and likely see quite a bit of rain. So make sure you have very good wet weather gear (preferably gortex or similar jacket, and water proof over-trousers), and good thermals and warm jumper/jacket.
I've done the overland track a few times, and it really is a lot of fun. I've done it once during winter with skis, and in some ways winter is better because there are fewer people, which suits me.